---
title: Soft-wall AdS/QCD Model Overview
url: https://www.emergentmind.com/topics/soft-wall-ads-qcd-model
type: topic
---

# Soft-wall AdS/QCD Model Overview

Soft-wall AdS/QCD is a bottom-up holographic framework in which the infrared sector of a five-dimensional asymptotically AdS description is regulated by a smooth dilaton background rather than a hard cutoff, with the principal aim of reproducing confinement-related features such as linear Regge trajectories while incorporating chiral symmetry breaking and other low-energy properties of QCD [1109.5523][1001.4799]. In standard formulations the geometry extends to \(z\to\infty\), but a factor such as \(e^{-\Phi(z)}\) suppresses the large-\(z\) region smoothly; in light-front-holographic formulations, a positive-sign dilaton metric yields an effective harmonic confining potential, a massless pion in the chiral limit, and linear trajectories in both radial and orbital quantum numbers [1009.4232][1007.5385]. Subsequent work recast the model as a five-dimensional rewriting of planar QCD sum-rule phenomenology, reconstructed scalar potentials supporting soft-wall backgrounds, constrained the infrared behavior by simultaneous meson and nucleon spectroscopy, and extended the framework to thermodynamics, dense matter, and chiral phase transitions [1001.3105][1105.6293][1909.06917].

## 1. Conceptual foundations

The starting point is usually AdS\(_5\) with metric
\[
ds^2=\frac{R^2}{z^2}\left(\eta_{\mu\nu}dx^\mu dx^\nu-dz^2\right),
\]
where the fifth-dimensional coordinate \(z\) plays the role of a holographic scale: small \(z\) corresponds to ultraviolet physics and large \(z\) to infrared physics [1009.4232][1007.5385]. In the soft-wall model, conformal invariance is broken smoothly by a dilaton profile rather than by a hard infrared cutoff. In one common convention, the action carries \(e^{-\Phi(z)}\) with \(\Phi(z)=c z^2\), so the large-\(z\) region is exponentially suppressed while the spacetime itself still extends to \(z\to\infty\) [1109.5523]. In another convention, especially in light-front holography, the soft wall is described through a positive-sign dilaton metric \(e^{+\kappa^2 z^2}\), which produces a confining potential with a minimum at \(z_0=1/\kappa\) and rising behavior at large \(z\) [1009.4232][1007.5385].

A central result of the light-front-holographic formulation is the identification \(z=\zeta\), where
\[
\zeta^2=x(1-x)\,\mathbf b_\perp^2
\]
is the invariant light-front transverse separation of constituents [1009.4232]. The corresponding light-front Schrödinger equation,
\[
\left( -\frac{d^2}{d\zeta^2} -\frac{1-4L^2}{4\zeta^2} +U(\zeta) \right)\phi(\zeta)=\mathcal M^2\phi(\zeta),
\]
with
\[
U(\zeta)=\kappa^4\zeta^2+2\kappa^2(L+S-1),
\]
gives the meson spectrum
\[
\mathcal M^2=4\kappa^2\left(n+L+\frac{S}{2}\right),
\]
so that \(\Delta \mathcal M^2=4\kappa^2\) for \(\Delta n=1\) and also for \(\Delta L=1\) [1009.4232]. For \(n=L=S=0\), one obtains \(\mathcal M_\pi^2=0\) in the massless-quark limit [1007.5385].

A complementary interpretation treats bottom-up holographic models as a five-dimensional rewriting of large-\(N_c\) QCD, where an infinite tower of narrow mesons in each channel is packaged into a single 5D field with a Sturm–Liouville spectrum [1001.3105]. Under mild assumptions, that analysis classifies the 5D backgrounds leading to simple Regge spectra and argues that the most phenomenologically consistent model is the soft wall, with a preference for the positive-sign dilaton background [1001.3105]. This suggests that the soft-wall model is not only an AdS/CFT-inspired construction but also an alternative language for planar QCD sum-rule phenomenology.

## 2. Bulk actions, background fields, and representative constructions

A standard meson-sector action is written in string frame as
\[
S_{\rm meson}=\int d^{5}x\sqrt{-g}\,e^{-\Phi(z)}\,\mathrm{Tr}\left[|DX|^{2}+m_{X}^{2}|X|^{2}+\kappa|X|^4+\frac{1}{4g_{5}^{2}\left(F_{L}^{2}+F_{R}^{2}\right)}\right],
\]
with bifundamental scalar \(X\), left and right gauge fields \(A_L,A_R\), and vacuum expectation value
\[
\langle X\rangle=\frac{\chi(z)}{2}I.
\]
The bulk gauge fields encode vector and axial-vector modes through
\[
F_{L,R}^{MN}=\partial^{M}A_{L,R}^{N}-\partial^{N}A_{L,R}^{M}-i[A_{L,R}^{M},A_{L,R}^{N}],
\qquad
D^{M}X=\partial^{M}X-iA_{L}^{M}X+iXA_{R}^{M}
\]
[1309.3823]. In improved meson models the scalar interaction is often quartic, and the same general structure is written as
\[
S_{5} = -\int d^{5}x\, \sqrt{-g}\, e^{-\chi(z)} \, \mathrm{Tr}\!\left[ |DX|^{2}+m_{X}^{2}|X|^{2} -\kappa |X|^{4} +\frac{1}{2 g_{5}^{2}\left(F_{A}^{2}+F_{V}^{2}\right) \right]
\]
or with \(z\)-dependent scalar mass and quartic coupling in pure AdS\(_5\) [1009.3009][1310.6487].

For the background sector, one frequently passes to Einstein frame. In one two-field form,
\[
S_{grav}=\frac{1}{16\pi G_5}\int d^{5}x\sqrt{-g_E}\left(R_E-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}\partial_\mu\chi\partial^\mu\chi-V(\phi,\chi)\right),
\]
and in the three-field extension the glueball condensate \(G\) is added as an additional scalar [1309.3823]. A systematic potential-reconstruction program shows how to build \(V(\phi,\chi)\) for prescribed soft-wall and chiral backgrounds, starting from the Einstein-frame action
\[
\mathcal{S}_E = \frac{1}{16 \pi G_5} \int d^{5}x \sqrt{-g_E} \left( R_E  -  \partial_\mu \phi \partial^\mu \phi -  \partial_\mu \chi \partial^\mu \chi - V(\phi,\chi) \right)
\]
and an exact AdS\(_5\) string-frame metric [1001.4799].

The main bottom-up variants represented in the literature can be organized succinctly as follows.

| Construction | Distinguishing ingredient | Representative papers |
|---|---|---|
| Standard soft wall | Quadratic dilaton and smooth IR suppression | [1109.5523], [1001.4799] |
| Light-front-holographic soft wall | Positive-sign dilaton metric and \(z=\zeta\) map | [1009.4232], [1007.5385] |
| Constrained IR background | \(\Phi\sim z^2\), \(a\sim z\), \(v\sim z^{-1}\) from mesons and nucleons | [1105.6293], [1205.0485] |
| Three-field background | Dilaton, chiral condensate, and glueball condensate \(G\) | [1309.3823], [1212.0044] |
| Improved thermal background | Einstein–dilaton geometry fitted to lattice thermodynamics | [1909.06917] |

These constructions share the same broad objective—smooth infrared breaking of conformal symmetry—but differ in how much of the background is imposed, reconstructed, or solved dynamically. A plausible implication is that “soft-wall AdS/QCD model” denotes a family of closely related bottom-up frameworks rather than a single universal action.

## 3. Chiral symmetry breaking and infrared engineering

A recurring difficulty of the original soft-wall model is that the combination of a purely quadratic scalar potential and the standard positive quadratic dilaton does not generate spontaneous chiral symmetry breaking. In the two-flavor analysis,
\[
\chi''+\left(3A_s'-\Phi'\right)\chi'+3e^{2A_s}\chi=0
\]
at zero temperature remains linear, and the exact \(T=0\) solution in the positive quadratic dilaton background gives
\[
\frac{\sigma(T=0)}{m_q\,\mu^2\zeta^2}=0.095,
\]
so \(\sigma\propto m_q\) and \(\sigma\to0\) in the chiral limit [1512.06493]. That work concludes that realistic chiral dynamics requires both a nonlinear scalar potential,
\[
V(\chi)=-\frac32\chi^2+v_3\chi^3+v_4\chi^4,
\]
and a dilaton profile that is negative in the UV/intermediate region but positive quadratic in the IR,
\[
\Phi(z)=-\mu_1^2z^2+(\mu_1^2+\mu_0^2)z^2\tanh(\mu_2^2z^2)
\]
[1512.06493]. In the two-flavor case this gives a second-order transition in the chiral limit and a crossover for any finite quark mass; in the three-flavor case the cubic term induced by the t’Hooft determinant interaction drives first-order behavior at small masses [1512.06493][1805.05019].

A different line of improvement keeps the geometry as pure \(AdS_5\) but moves the necessary infrared deformation into the matter sector. The infrared-improved soft-wall model for mesons uses
\[
\Phi(z)=\mu_g^2 z^2-\frac{\lambda_g^4\mu_g^4 z^4}{(1+\mu_g^2 z^2)^3},
\]
a \(z\)-dependent quartic coupling
\[
\lambda_X(z)=\frac{\mu_g^2 z^2}{1+\mu_g^2 z^2}\lambda,
\]
and a \(z\)-dependent scalar mass \(m_X^2(z)\) whose UV limit approaches \(-3\) and whose IR asymptotics are fixed by the scalar equation of motion [1310.6487]. The scalar VEV is parameterized as
\[
v(z)=\frac{A z+B z^3}{1+C z^2},
\]
with UV source/VEV behavior and linear IR growth [1310.6487]. This arrangement is designed to realize linear confinement and chiral symmetry breaking within one simple framework.

A further constraint arises when mesons and nucleons are fitted simultaneously. Requiring linear Regge behavior in both sectors and equal asymptotic slopes for vector and axial-vector mesons leads to the infrared conditions
\[
\Phi(z)\sim z^2,\qquad a(z)\sim z,\qquad v(z)\sim z^{-1}\qquad (z\to\infty),
\]
while the UV remains
\[
a(z)\sim \frac{L}{z},\qquad v(z)\sim A z + B z^3,\qquad m_X^2(z)\to -3
\]
[1105.6293]. These asymptotics motivate the rational interpolating forms
\[
a(z)=\frac{1+\mu z^2}{z},\qquad v(z)=\frac{A z + B z^3}{1 + C z^4}
\]
used later in the quartic scalar model of mesons and nucleons [1105.6293][1205.0485].

The most explicit structural critique of two-field soft-wall backgrounds appears in the three-field construction. In the two-field case one has the potential-independent relation
\[
6a\,\phi''(z)+[\phi'(z)]^{2}(6a^{2}-1)-[\chi'(z)]^{2}+\frac{12a\,\phi'(z)}{z}=0,
\]
and imposing the soft-wall infrared forms
\[
\phi(z)\sim \lambda z^2,\qquad \chi(z)\sim \Gamma z
\]
forces
\[
\Delta m^2 = \lim_{z\rightarrow\infty} \frac{g_5^2\chi^2}{z^2}=g_5^2\Gamma^2=g_5^2 6^{3/2}\lambda,
\]
which predicts a vector/axial-vector mass splitting about \(50\) times too large [1309.3823]. Introducing a glueball condensate \(G(z)\), dual to the gluonic operator \(\mathrm{Tr}\,F^2\), changes the key background equation to
\[
\chi'^2+G'^2=\frac{\sqrt{6}}{z^{2}}\frac{d}{dz}\left(z^{2}\phi'\right),
\]
so the chiral field no longer bears the full burden of sourcing the quadratic dilaton [1309.3823]. The UV and IR asymptotics are then
\[
\chi(z)\sim \sigma z^3,\qquad G(z)\sim g_o z^4\qquad (z\to0),
\]
and
\[
\chi(z)\sim \Gamma z,\qquad G(z)\sim z,\qquad \phi(z)\sim \lambda z^2\qquad (z\to\infty)
\]
[1309.3823]. The authors explicitly state that constructing a globally valid scalar potential consistent with both UV and IR limits remains open, but the three-field parametrization resolves the major phenomenological inconsistency of the two-field setup [1309.3823].

## 4. Spectroscopy of mesons and baryons

The original phenomenological appeal of the soft wall lies in spectroscopy. In the light-front-holographic version, the meson mass formula
\[
\mathcal M^2=4\kappa^2\left(n+L+\frac{S}{2}\right)
\]
gives linear trajectories with equal slopes in the radial quantum number \(n\) and orbital angular momentum \(L\), and the baryon sector satisfies
\[
\mathcal M^2=4\kappa^2(n+\nu+1),\qquad \nu=L+1
\]
[1009.4232]. The same framework yields explicit proton wavefunctions,
\[
\psi_+(\zeta)=\sqrt{2}\,\kappa^2\zeta^{3/2}e^{-\kappa^2\zeta^2/2}, \qquad
\psi_-(\zeta)=\kappa^3\,\zeta^{5/2}e^{-\kappa^2\zeta^2/2},
\]
and predicts a massless pion for zero quark mass [1009.4232][1007.5385].

In meson-sector bottom-up models, the fluctuation equations are typically cast into Sturm–Liouville or Schrödinger-like form. The vector sector depends on the dilaton and warp factor, while the axial sector contains an additional effective mass term proportional to \(\chi^2/z^2\), which is why the infrared behavior of the chiral background is crucial [1309.3823]. The constrained-IR model with quartic scalar interaction reports good agreement with experimental meson and nucleon spectra once the background functions are chosen as
\[
\Phi(z)=\kappa^2 z^2,\qquad a(z)=\frac{1+\mu z^2}{z},\qquad v(z)=\frac{A z + B z^3}{1 + C z^4}
\]
and a running bulk spinor mass is introduced for the nucleons [1205.0485].

The pseudoscalar sector required additional clarification. In the modified soft-wall model with quartic scalar interaction and a nontrivial scalar vacuum profile \(v(z)\), the pion can be represented either exponentially,
\[
X_e = \left(\frac{v(z)}{2}+S(x,z)\right) I \, e^{2i\pi_e^a(x,z)t^a},
\]
or linearly,
\[
X_l = \left(\frac{v(z)}{2}+S(x,z)\right) I + i\pi_l^a(x,z)t^a.
\]
The two representations are related by
\[
\pi_l=v(z)\,\pi_e,
\]
and become exactly equivalent once the scalar background equation is used [1009.3009]. Solving the full coupled system, rather than reducing it prematurely to a single equation for \(\partial_z\pi\), produces a realistic pion tower with a very light ground state, a large gap to the first radial excitation, and a roughly linear radial trajectory [1009.3009]. The Gell-Mann–Oakes–Renner relation,
\[
2m_q \sigma = m_\pi^2 f_\pi^2,
\]
also follows naturally [1009.3009].

Infrared-improved meson models sharpen these results. One such model reports scalar, pseudoscalar, vector, and axial-vector masses
\[
460,\ 1359,\ 1723,\ 1989,\ 2216,\ 2417,\ 2602\ {\rm MeV}
\]
for the scalar tower,
\[
775.5,\ 1354,\ 1667,\ 1919,\ 2140,\ 2339,\ 2523\ {\rm MeV}
\]
for vectors, and
\[
1095,\ 1626,\ 1913,\ 2145,\ 2350,\ 2537,\ 2709\ {\rm MeV}
\]
for axial-vectors, with the lightest scalar around \(460\) MeV and a reasonable space-like pion form factor [1310.6487]. In the three-field soft-wall background, a least-squares fit to the \(\rho\) and \(a_1\) spectra gives
\[
\sigma = (0.375\,{\rm GeV})^3,\qquad g_o=(1.5\,{\rm GeV})^4,\qquad \Gamma = 0.25\,{\rm GeV},\qquad \lambda = (0.428\,{\rm GeV})^2,
\]
and the quartic coupling
\[
\kappa = 12.4,
\]
which removes the unbounded-from-below scalar potential of earlier work and avoids a virtual \(f_0\) ground state [1309.3823].

Not all observables are reproduced equally well in the standard prescription. For vector-meson decay constants, the original soft wall predicts
\[
m_n^2 = 4kn
\]
and a radial-independent decay constant
\[
f_n = \frac{k^{1/2}}{\tilde g_5}\sqrt{2},
\]
which is inconsistent with heavy-quarkonium data, where the decay constants decrease with excitation number [1507.04708]. A modified prescription evaluates the current-current correlator at a finite radial position \(z=z_0\), with bulk-to-boundary propagator
\[
v(p,z)= \frac{U\!\left(\frac{p^2}{4k},\,0,\,k z^2\right)}
{U\!\left(\frac{p^2}{4k},\,0,\,k z_0^2\right)},
\]
thereby introducing a second ultraviolet scale \(1/z_0\) and yielding decreasing decay constants for heavy vector mesons [1507.04708].

## 5. Thermal, dense, and phase-structure extensions

Finite-temperature and finite-density soft-wall models typically introduce a black-hole background and a bulk \(U(1)\) field dual to quark number. In one standard treatment, the gravity action is
\[
S_{\text{grav}}=-\frac{1}{2\kappa^2}\int d^5x\,\sqrt{g}\,e^{-\Phi}\left(R+\frac{12}{L^2}\right),
\]
with
\[
\Phi(z)=c z^2,
\]
thermal AdS as the confined phase,
\[
ds^2=\frac{L^2}{z^2}\left(d\tau^2+dz^2+d\vec{x}^2\right),
\]
and AdS black hole as the deconfined phase,
\[
ds^2=\frac{L^2}{z^2}\left(f(z)d\tau^2+\frac{dz^2}{f(z)}+d\vec{x}^2\right),\qquad
f(z)=1-\left(\frac{z}{z_h}\right)^4,\qquad
T=\frac{1}{\pi z_h}
\]
[1109.5523]. At \(\mu=0\), the Hawking–Page-type transition occurs at
\[
c z_h^2=0.419,\qquad T_c = 0.492\,\sqrt{c},
\]
and the quark number susceptibility is
\[
\chi(T) = \frac{c\,N_cN_f}{2\pi^2\left(e^{c/\pi^2 T^2}-1\right)},
\]
which reduces at high temperature to
\[
\chi(T)\approx \frac{N_cN_f T^2}{2}
\]
[1109.5523]. That paper explicitly notes that the results are valid only for small chemical potential because backreaction is neglected [1109.5523].

The phase boundary can also be modeled spectrally rather than thermodynamically. In an extended soft-wall model, vector mesons are identified with normalizable bulk modes, and their disappearance at \(T_{\mathrm{dis}}(\mu)\) is adjusted to track the freeze-out or crossover curve at small \(\mu\) [1611.04124]. The background metric is
\[
d s^2 =  e^{A(z)} \left( f(z) d t^2 - d \vec x ^{\, 2} -\frac{1}{f(z)} d z^2 \right),
\]
with generalized dilaton
\[
\Phi(z) = (cz)^p
\]
and a phenomenological blackness function engineered to reproduce thermal-gas versus black-hole competition [1611.04124]. A plausible implication is that in soft-wall practice the deconfinement line can be defined either from free-energy competition or from the disappearance of normalizable hadronic modes, depending on the construction.

Chiral transitions require a separate scalar-sector analysis. In the improved soft-wall model with nonlinear scalar potential and interpolating dilaton, the two-flavor case exhibits a second-order transition in the chiral limit and a crossover for finite quark mass, while the three-flavor case exhibits first-order behavior in the chiral limit and at small finite masses, with a critical mass separating first-order and crossover regimes [1512.06493]. The \(2+1\)-flavor extension uses
\[
m_5^2(z)=-3-\mu_c^2 z^2,
\qquad
\Phi(z)=\mu_g^2 z^2,
\]
plus a determinant term \(\gamma\det[X]\), and yields a Columbia-plot structure with a tricritical point on the \(m_{u,d}=0\) boundary at
\[
m_s^{\rm tri}\simeq 212.7~\mathrm{MeV}
\]
[1805.05019].

A more realistic thermal background is obtained by solving an Einstein–dilaton system with nontrivial dilaton potential, rather than imposing AdS-Schwarzschild. In that model the string-frame metric is
\[
ds^2 = \frac{L^2 e^{2 A_S(z)}}{z^2} \left(-f(z)dt^2 + dx^i dx^i +\frac{dz^2}{f(z)}\right),
\]
the Einstein-frame action is
\[
S_g =\frac{1}{2\kappa_5^2}\int d^5x\sqrt{-g_E}\left[R_E -\frac{4}{3}(\partial\phi)^2 -V_E(\phi)\right],
\]
and the potential
\[
V_c(\phi_c) =\frac{1}{L^2}\left(-12\cosh\gamma\phi_c +b_2\phi_c^2 +b_4\phi_c^4\right)
\]
is tuned to reproduce the two-flavor lattice equation of state [1909.06917]. On top of that background, an improved soft-wall flavor action with dilaton–scalar coupling
\[
V_X(X,\Phi) =m_5^2|X|^{2} -\lambda_1\Phi |X|^{2} +\lambda_2|X|^{4}
\]
produces a crossover chiral transition, although the chiral transition temperature remains too high relative to lattice [1909.06917].

Finite isospin density exposes additional limitations. In a soft-wall pion-condensation study with improved dilaton
\[
\Phi(z)=-\mu_1^2 z^2+(\mu_1^2+\mu_0^2)z^2\tanh(\mu_2^2 z^2)
\]
and AdS-Reissner–Nordström-like background,
\[
f(z)=1-(1+\gamma \mu_I^2 z_h^2)\frac{z^4}{z_h^4}+\gamma \mu_I^2\frac{z^6}{z_h^4},
\qquad
V_0^3(z)=\mu_I\left(1-\frac{z^2}{z_h^2}\right),
\]
the phase transition into and out of the pion-condensed phase is found to be first order both at small and large \(\mu_I\) [1811.03828]. The same paper states that the disagreement with lattice behavior at small \(\mu_I\) indicates that a full back-reaction model including the interaction of gluodynamics and chiral dynamics is likely necessary [1811.03828].

## 6. Conventions, controversies, and open problems

A persistent source of confusion in the literature is the sign convention for the dilaton. In the thermal soft-wall model and many meson-sector constructions, the action carries \(e^{-\Phi}\) with \(\Phi=cz^2\) [1109.5523]. In the light-front-holographic formulation, by contrast, the defining statement is that the confining choice is a positive-sign dilaton metric,
\[
\varphi(z)=+\kappa^2 z^2,
\]
opposite in sign to the original Karch–Katz–Son–Stephanov convention [1009.4232][1007.5385]. Afonin’s classification of five-dimensional large-\(N_c\) rewritings also argues for a preference for the positive-sign dilaton background [1001.3105]. These are not merely superficial notational differences: they arise in different frames and different effective formulations, and each comes with its own interpretation of the confining background.

Another recurring issue is the bottom-up status of the framework. Several papers emphasize that soft-wall AdS/QCD is phenomenological rather than derived from a full top-down string construction [1109.5523][1512.06493][1909.06917]. The large-\(N_c\) rewriting program goes further and argues that many features of AdS/QCD can be understood without invoking the full AdS/CFT correspondence at all, as a 5D reformulation of planar QCD sum rules [1001.3105]. This suggests that the success of the soft wall may depend more on its spectral and symmetry structure than on a strict microscopic duality.

Dynamic AdS/QCD sharpens the question of what a soft wall means physically. In that framework the running constituent mass \(L(\rho)\) itself generates an infrared wall through
\[
r^2=\rho^2+L(\rho)^2,
\]
so if \(L(0)\neq0\), meson dynamics never reaches \(r=0\) [1508.06540]. The natural outcome is then square-well-like behavior,
\[
M^2\sim n^2,\quad M^2\sim s^2,
\]
rather than Regge behavior. To obtain
\[
M_{n,s}^2 \sim n,\ s,
\]
one must force \(L(\rho)\) to fall sharply in the IR, for example by taking
\[
L = \frac{\rho}{1 + \rho^3}, \qquad \Phi = z^2,
\]
or by other engineered profiles [1508.06540]. That paper concludes that Regge-producing soft-wall behavior requires meson physics to remain sensitive to scales below the quark on-shell mass, and therefore that the meson spectrum is highly sensitive to IR decoupling [1508.06540]. A plausible implication is that “soft-wall behavior” and “dynamical constituent-mass screening” are not automatically compatible.

Several open problems remain explicit in the literature. The three-field model resolves the structural two-field obstruction but does not yet supply a globally valid scalar potential consistent with both UV and IR limits [1309.3823]. The pion-condensation study indicates that full backreaction may be required to capture low-\(\mu_I\) physics correctly [1811.03828]. The Einstein–dilaton thermal model shows that deconfinement-related thermodynamics and chiral restoration can be treated in one framework, but still in the probe approximation for the flavor sector [1909.06917]. Across the literature, the soft-wall AdS/QCD model therefore appears less as a finished theory than as an evolving family of constrained effective constructions whose defining achievement is the simultaneous modeling of smooth infrared confinement, Regge behavior, and chiral dynamics within a five-dimensional holographic language [1001.4799][1309.3823].

Source: https://www.emergentmind.com/topics/soft-wall-ads-qcd-model