Hard-Wall and Soft-Wall Models

The main idea

Pure AdS is too conformal to describe ordinary low-energy QCD. It has no intrinsic mass scale, no discrete tower of massive hadrons, and no confining flux tube in the Wilson-loop sense. A bottom-up AdS/QCD model therefore begins with a pragmatic question:

What is the minimal five-dimensional structure that preserves the ultraviolet logic of the AdS/CFT dictionary but modifies the infrared enough to model confinement and hadron physics?

The two classic answers are the hard wall and the soft wall.

The hard-wall model keeps the AdS$_5$ metric

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad 0<z\le z_m,$$

and simply cuts off the radial direction at

$$z_m\sim \Lambda_{\mathrm{QCD}}^{-1}.$$

The cutoff acts like an infrared box. Bulk normal modes become discrete, with masses roughly set by zeros of Bessel functions. For many hadron towers, this gives

$$m_n\sim \frac{n}{z_m}, \qquad m_n^2\sim n^2.$$

The soft-wall model removes the abrupt wall and instead introduces a smooth infrared profile, often written as a dilaton factor in the five-dimensional action,

$$S\sim \int d^5x\sqrt{-g}\,e^{-\Phi(z)}\mathcal L, \qquad \Phi(z)=\kappa^2 z^2.$$

This changes the radial Sturm-Liouville problem into something close to a harmonic oscillator. For vector mesons in the simplest model one finds

$$m_n^2=4\kappa^2(n+1), \qquad n=0,1,2,\ldots,$$

which gives the linear radial Regge behavior

$$m_n^2\sim n.$$

The hard wall introduces confinement by ending the AdS radial interval at $z=z_m$. The soft wall replaces the sharp cutoff by a smooth infrared profile, commonly $\Phi(z)=\kappa^2z^2$, producing oscillator-like radial spectra and $m_n^2\sim n$.

These models are extremely useful, but their status must be stated honestly. They are not known to be exact duals of large-$N_c$ QCD. They are phenomenological five-dimensional effective models inspired by AdS/CFT. Their strength is that they package large-$N_c$ counting, current correlators, chiral symmetry, confinement scales, and hadron spectra into a calculable geometric framework. Their weakness is that the infrared geometry and field content are chosen by modeling judgment, not derived from a complete string construction of QCD.

A good way to use them is this:

$$\text{AdS/CFT dictionary} \quad+ \text{QCD symmetries} \quad+ \text{large-}N_c\text{ counting} \quad+ \text{phenomenological IR input}.$$

A bad way to use them is to say: “this is QCD because it is five-dimensional.” Tiny sentence, enormous crime scene.

What problem are these models solving?

A four-dimensional confining gauge theory has a mass gap and a discrete spectrum of color-singlet hadrons. In the large-$N_c$ limit, stable mesons and glueballs have parametrically narrow widths. For example, normalized single-trace meson operators have two-point functions with poles

$$\langle J_\mu(q)J_\nu(-q)\rangle \sim \sum_{n=0}^{\infty} \frac{f_n^2}{q^2+m_n^2} \left(\eta_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}\right) + \text{contact terms}$$

in Euclidean signature. At large $N_c$, the widths scale as

$$\Gamma_n\sim \frac{1}{N_c},$$

so the spectral density becomes a sum over narrow resonances.

Pure Poincaré AdS does not naturally produce such a spectrum. A free scalar in Euclidean AdS$_5$ with momentum $q$ satisfies a radial equation whose normalizable solutions behave continuously in $q^2$. This is appropriate for a CFT, whose two-point functions have branch cuts rather than isolated hadron poles. To model QCD-like physics, we need to modify the infrared.

The minimal requirements are:

QCD feature	Holographic modeling ingredient
approximate UV scaling	asymptotically AdS metric near $z=0$
confinement scale	IR wall, cap, or smooth dilaton/warp-factor scale
discrete hadrons	normalizable radial eigenmodes
large-$N_c$ narrow resonances	classical five-dimensional fields
flavor symmetry	five-dimensional gauge fields for $SU(N_f)_L\times SU(N_f)_R$
chiral symmetry breaking	bifundamental scalar or brane embedding data
current correlators	on-shell action differentiated with respect to UV sources

The hard wall and soft wall are the two simplest laboratories for this list.

The bottom-up dictionary

In a top-down string construction, the bulk fields, compact space, interactions, and boundary conditions are fixed by the higher-dimensional theory. In bottom-up AdS/QCD, one instead chooses a small set of five-dimensional fields dual to important QCD operators.

For light-flavor mesons one often introduces five-dimensional gauge fields

$$L_M=L_M^a t^a, \qquad R_M=R_M^a t^a,$$

dual to the chiral currents

$$J_{L}^{\mu,a}=\bar q_L\gamma^\mu t^a q_L, \qquad J_{R}^{\mu,a}=\bar q_R\gamma^\mu t^a q_R.$$

It is convenient to define vector and axial combinations

$$V_M=\frac12(L_M+R_M), \qquad A_M=\frac12(L_M-R_M).$$

One also introduces a bifundamental scalar

$$X\sim \bar q_R q_L,$$

transforming as

$$X\to U_L X U_R^\dagger$$

under $SU(N_f)_L\times SU(N_f)_R$. Since $\bar q q$ has engineering dimension $\Delta=3$, the AdS mass-dimension relation in $d=4$ gives

$$m_X^2L^2=\Delta(\Delta-4)=-3.$$

Near the boundary,

$$X(z) \sim \frac12\left(m_q z+\sigma z^3\right),$$

where $m_q$ is the quark-mass source and $\sigma$ is proportional to the chiral condensate. The factor $1/2$ is conventional.

The simplest hard-wall meson action is schematically

$$S = \int d^5x\sqrt{-g}\,\mathrm{Tr} \left[ |D X|^2+m_X^2|X|^2 - \frac{1}{4g_5^2}\left(F_L^2+F_R^2\right) \right],$$

with the fields living on the interval $0<z\le z_m$.

The five-dimensional gauge coupling $g_5$ is not arbitrary if one wants the ultraviolet current-current correlator to match perturbative QCD at large Euclidean momentum. The standard matching gives

$$g_5^2=\frac{12\pi^2}{N_c}$$

in common normalization. This is a good example of the bottom-up philosophy: the UV part of the model is fixed by operator matching, while the IR wall and some boundary conditions are phenomenological.

The hard-wall vector spectrum

Let us see the main mechanism explicitly. Consider a five-dimensional Maxwell field in AdS$_5$ on the hard-wall interval. The quadratic action is

$$S_V = - \frac{1}{4g_5^2} \int d^5x\sqrt{-g}\,F_{MN}F^{MN}.$$

Choose radial gauge

$$V_z=0,$$

and transverse four-dimensional modes

$$\partial^\mu V_\mu=0.$$

Write

$$V_\mu(x,z)=\epsilon_\mu v_n(z)e^{ip\cdot x}, \qquad p^2=-m_n^2.$$

The radial equation becomes

$$\partial_z\left(\frac{1}{z}\partial_z v_n\right) + \frac{m_n^2}{z}v_n=0.$$

The solution regular near the UV boundary is

$$v_n(z)=C_n z J_1(m_n z).$$

A common IR condition for vector mesons is Neumann,

$$\partial_z v_n(z_m)=0.$$

Using

$$\frac{d}{dz}\left[zJ_1(mz)\right]=mzJ_0(mz),$$

we get

$$J_0(m_n z_m)=0.$$

Therefore

$$m_n=\frac{\gamma_{0,n}}{z_m},$$

where $\gamma_{0,n}$ is the $n$th zero of $J_0$. For large $n$,

$$\gamma_{0,n}\sim \left(n-\frac14\right)\pi,$$

so

$$m_n\sim \frac{\pi n}{z_m}, \qquad m_n^2\sim n^2.$$

This is the central success and central limitation of the hard wall. It naturally gives a discrete spectrum and a mass gap. But highly excited QCD mesons are better organized by approximately linear Regge behavior,

$$m_{n,S}^2\sim n+S,$$

not by $m_n^2\sim n^2$.

Hard-wall correlators and poles

The same equation also computes the vector-current two-point function. Instead of a normalizable mode, solve the bulk-to-boundary problem

$$V_\mu(q,z)=V_\mu^{(0)}(q)\,\mathcal V(q,z), \qquad \mathcal V(q,0)=1.$$

The on-shell Maxwell action reduces to a boundary term,

$$S_{\mathrm{os}} = - \frac{1}{2g_5^2} \int\frac{d^4q}{(2\pi)^4} V_\mu^{(0)}(-q)V^{(0)\mu}(q) \left.\frac{1}{z}\mathcal V(-q,z)\partial_z\mathcal V(q,z)\right|_{z=\epsilon}^{z=z_m}.$$

After imposing the IR condition and renormalizing the UV divergence, the current correlator is obtained by differentiating with respect to $V_\mu^{(0)}$. The normalizable modes appear as poles in the two-point function. This is the same GKPW logic as before, but with an IR boundary condition that discretizes the spectrum.

In this sense the hard wall is a five-dimensional version of a large-$N_c$ resonance model. The UV source creates a current insertion; the current couples to a tower of normalizable bulk modes; those modes are interpreted as mesons.

Chiral symmetry breaking in the hard wall

The hard-wall model can also describe the basic chiral symmetry pattern

$$SU(N_f)_L\times SU(N_f)_R \longrightarrow SU(N_f)_V.$$

The bifundamental scalar has a background profile

$$X(z)=\frac12 v(z)\mathbf 1_{N_f},$$

with

$$v(z)=m_q z+\sigma z^3$$

in the simplest hard-wall setup. The nonzero $v(z)$ gives a $z$-dependent mass to the axial gauge field through $|DX|^2$, splitting vector and axial-vector mesons. Pions arise from the coupled system involving the phase of $X$ and the longitudinal part of $A_M$.

A famous success of the original hard-wall construction is that it naturally reproduces the Gell-Mann-Oakes-Renner relation,

$$f_\pi^2m_\pi^2=2m_q\sigma,$$

up to conventions for the normalization of $\sigma$.

There is an important conceptual caveat. In the simplest hard-wall model, $m_q$ and $\sigma$ are treated as independent input parameters because the scalar equation in fixed AdS permits both coefficients. In real QCD, the condensate is dynamically determined once the quark masses and gauge dynamics are specified. More refined holographic models try to improve this by adding scalar potentials, backreaction, tachyon dynamics, or top-down brane embeddings.

Why soft walls were introduced

The hard wall is brutally effective: it creates a mass gap by ending space. But an abrupt radial cutoff has two aesthetic and physical drawbacks.

First, the large excitation spectrum is box-like:

$$m_n^2\sim n^2.$$

Second, the wall is not generated dynamically by a smooth solution of bulk equations. It is imposed by hand.

The soft-wall idea is to replace the sharp cutoff by a smooth infrared profile that suppresses wavefunctions at large $z$. The simplest implementation keeps the AdS metric but modifies the action by

$$e^{-\Phi(z)}, \qquad \Phi(z)=\kappa^2z^2.$$

For a vector field,

$$S_V = - \frac{1}{4g_5^2} \int d^5x\sqrt{-g}\,e^{-\Phi(z)}F_{MN}F^{MN}.$$

The mode equation becomes

$$\partial_z\left(\frac{e^{-\Phi}}{z}\partial_z v_n\right) + \frac{m_n^2e^{-\Phi}}{z}v_n=0.$$

This is again a Sturm-Liouville problem, but now on the half-line $0<z<\infty$. The infrared boundary condition is normalizability with respect to the weight $e^{-\Phi}/z$, not an imposed condition at $z_m$.

The soft-wall spectrum as a harmonic oscillator

The vector equation can be put in Schrödinger form. Write

$$\partial_z\left(e^{-B}\partial_z v_n\right) + m_n^2e^{-B}v_n=0, \qquad B(z)=\Phi(z)+\log z.$$

Now set

$$v_n(z)=e^{B(z)/2}\psi_n(z).$$

Then

$$-\psi_n''+U(z)\psi_n=m_n^2\psi_n,$$

where

$$U(z)=\frac{B'(z)^2}{4}-\frac{B''(z)}{2}.$$

For

$$\Phi(z)=\kappa^2z^2, \qquad B(z)=\kappa^2z^2+\log z,$$

we find

$$B'(z)=2\kappa^2 z+\frac1z, \qquad B''(z)=2\kappa^2-\frac1{z^2}.$$

Thus

$$U(z) = \kappa^4z^2+\frac{3}{4z^2}.$$

This is a radial harmonic-oscillator potential. The normalizable spectrum is

$$m_n^2=4\kappa^2(n+1), \qquad n=0,1,2,\ldots.$$

The eigenfunctions may be written as

$$v_n(z)\propto z^2 L_n^1(\kappa^2z^2),$$

with the appropriate exponential factor appearing in the Schrödinger wavefunction $\psi_n$.

The result is the main reason the soft wall became popular: it gives linear radial trajectories without a hard IR brane.

Regge behavior and spin

A basic empirical feature of hadron spectroscopy is approximate Regge behavior,

$$m_{n,S}^2\sim n+S,$$

where $n$ is the radial excitation number and $S$ is the spin. In a semiclassical string picture, this is natural because the mass squared of a rotating string grows linearly with angular momentum.

A hard-wall model is not well adapted to this behavior at high excitation number. A soft wall with quadratic dilaton is better: the radial equation becomes oscillator-like, producing linear $m^2$ trajectories. In its simplest version, one often quotes

$$m_{n,S}^2\sim 4\kappa^2(n+S+\text{intercept}),$$

where the intercept depends on the field, spin, and model details.

This formula should not be oversold. The soft wall is a phenomenological way to encode the desired large-excitation behavior. In a complete confining string dual, the Regge physics would come from a full string background and its worldsheet dynamics. The soft wall captures one important spectral feature with a very economical five-dimensional model.

Hard wall versus soft wall

The two models are best viewed as complementary approximations.

Feature	Hard wall	Soft wall
IR scale	sharp cutoff $z=z_m$	smooth profile $\Phi\sim \kappa^2z^2$
radial domain	finite interval	half-line
normal modes	box-like Bessel modes	oscillator-like modes
large-$n$ vector spectrum	$m_n^2\sim n^2$	$m_n^2\sim n$
simplest calculation	boundary-value problem with IR condition	Sturm-Liouville problem with normalizability
confinement intuition	space ends in the IR	wavefunctions are suppressed in the IR
main virtue	simple, concrete, good for low-lying modes	linear Regge trajectories
main drawback	abrupt nondynamical wall	dilaton/background often imposed by hand

The hard wall is often easier for teaching because boundary conditions are visible. The soft wall is often more realistic for high radial excitations because it produces linear trajectories. Neither should be mistaken for a controlled top-down dual of QCD.

What does “confinement” mean in these models?

The word confinement has several related but distinct meanings:

1. A mass gap.
2. A discrete color-singlet spectrum.
3. An area law for large Wilson loops.
4. Vanishing expectation value of the Polyakov loop in a confining phase.
5. Absence of colored asymptotic states.
6. Linear Regge trajectories.

Hard-wall and soft-wall models can capture some of these features but not all automatically.

The hard wall produces a mass gap and discrete normal modes. It can also produce an area-law-like Wilson-loop behavior if the string worldsheet is forced to sit near the IR wall at large separation. But the wall itself is imposed, so the model does not dynamically explain confinement.

The soft wall produces a mass gap and linear radial spectra for suitable fields. But a dilaton factor in a probe-field action does not automatically guarantee Wilson-loop confinement. For Wilson loops, the relevant object is the string-frame geometry seen by a fundamental string. A background that confines meson wavefunctions in a five-dimensional action may or may not produce an area law for a fundamental string.

This distinction matters. Spectral confinement and Wilson-loop confinement are related in genuine confining gauge theories, but a bottom-up model can realize one more cleanly than the other.

The Wilson-loop criterion in geometric language

For a static quark-antiquark pair, the holographic Wilson-loop prescription uses the Nambu-Goto action of a fundamental string. In a general string-frame metric of the form

$$ds_s^2 = g_{tt}(z)dt^2+g_{xx}(z)d\vec x^{\,2}+g_{zz}(z)dz^2+\cdots,$$

the string tension felt at radial position $z$ is controlled by

$$T_{\mathrm{eff}}(z) = \frac{1}{2\pi\alpha'}\sqrt{g_{tt}(z)g_{xx}(z)}.$$

A sufficient intuitive condition for confinement is that $T_{\mathrm{eff}}(z)$ has a positive minimum in the infrared, so a large string worldsheet prefers to lie at that radial position. Then

$$V(R)\sim \sigma_{\mathrm{string}}R, \qquad \sigma_{\mathrm{string}} = \min_z T_{\mathrm{eff}}(z).$$

In a hard-wall model, the minimum is effectively at the wall. In a smooth top-down confining geometry, it may occur where a circle caps off or where the string-frame warp factor has a minimum. In a soft-wall meson model with only a dilaton in the field action, one must ask separately what the string-frame geometry is. This is a recurring theme: different observables probe different parts of the would-be dual.

How to compute in hard-wall or soft-wall models

A typical bottom-up computation follows a standard pattern.

1. Choose the operator

For a QCD operator $\mathcal O$ of dimension $\Delta$ and spin $s$, choose a five-dimensional field $\Phi_{\mathcal O}$ with the corresponding UV behavior. For a scalar in AdS$_5$,

$$m_5^2L^2=\Delta(\Delta-4).$$

For a conserved current, use a five-dimensional gauge field. For the stress tensor or glueball sectors, one may use metric or scalar-gravity fluctuations, depending on the model.

2. Choose the background

The simplest backgrounds are:

$$\text{hard wall:} \qquad 0<z\le z_m, \qquad \Phi=0,$$

and

$$\text{soft wall:} \qquad 0<z<\infty, \qquad \Phi=\kappa^2z^2.$$

More sophisticated models replace this by an Einstein-dilaton background,

$$ds^2=e^{2A(z)}(dz^2+dx_{1,3}^2), \qquad \Phi=\Phi(z),$$

where $A(z)$ and $\Phi(z)$ are solved from equations of motion or fitted to thermodynamics and spectra.

3. Solve the radial eigenvalue problem

A general mode equation has Sturm-Liouville form

$$-\partial_z\left[p(z)\partial_z\psi_n(z)\right]+q(z)\psi_n(z) =m_n^2w(z)\psi_n(z),$$

with normalization

$$\int dz\,w(z)\psi_m(z)\psi_n(z)=\delta_{mn}.$$

The masses $m_n$ are eigenvalues. The wavefunctions determine decay constants and cubic couplings through overlap integrals.

4. Compute correlators from the on-shell action

For two-point functions, solve the non-normalizable problem with source normalization at the boundary. The renormalized on-shell action gives

$$\langle \mathcal O(q)\mathcal O(-q)\rangle = \frac{\delta^2 S_{\mathrm{ren}}}{\delta J(q)\delta J(-q)}.$$

The poles of this correlator occur at the normal-mode eigenvalues.

5. Interpret the answer with model discipline

Ask three questions before believing the result:

• Was the UV normalization matched to QCD or chosen freely?
• Is the IR scale fixed from one observable and then used to predict another?
• Is the observable sensitive to the imposed wall/dilaton in a way that could change in a different model?

This discipline separates useful phenomenology from decorative holography.

A useful example: vector meson dominance

Because the five-dimensional gauge field is dual to the flavor current, the current correlator naturally decomposes into normal modes:

$$\Pi_V(q^2) \sim \sum_n\frac{f_{V,n}^2}{q^2+m_{V,n}^2}.$$

This resembles vector meson dominance: external current sources couple to a tower of vector resonances. In hard-wall and soft-wall models, the residues $f_{V,n}$ are determined by the UV behavior of the normalizable wavefunctions. Schematically,

$$f_{V,n} \propto \left.\frac{1}{g_5}\frac{\partial_z v_n(z)}{z}\right|_{z=0}.$$

This is a nice example of what bottom-up holography does well. It turns a qualitative large-$N_c$ statement — current correlators are sums over narrow resonances — into a calculable radial eigenvalue problem.

Relation to top-down confinement

Hard-wall and soft-wall models are not the only way to obtain confinement holographically. In more geometric models, the infrared can arise from a smooth cap, wrapped branes, fluxes, or a running dilaton. Examples include Witten’s D4-brane model, Klebanov-Strassler-type cascading geometries, and other top-down or semi-top-down constructions.

A smooth confining geometry often has one of the following features:

$$\text{IR circle caps off smoothly}, \qquad \text{warp factor has a minimum}, \qquad \text{fluxes generate a mass gap}, \qquad \text{compact space degenerates regularly}.$$

The bottom-up hard wall mimics the existence of an IR endpoint. The soft wall mimics a smooth confining potential in the radial wave equation. They are useful because they are analytically simple, not because they replace the problem of deriving QCD from string theory.

Common mistakes

Mistake 1: calling the hard wall “the dual of QCD”

The hard wall is a model inspired by holography. It is not a derived string dual of QCD. It has too few fields, no asymptotic freedom, and an imposed IR cutoff.

Mistake 2: confusing a mass gap with full confinement

A discrete spectrum is not the same as proving Wilson-loop area law, chiral symmetry breaking, center symmetry, and absence of colored states. Holographic models can capture different confinement diagnostics with different degrees of reliability.

Mistake 3: using the wrong spectrum at high excitation

Hard-wall spectra behave like a box at large radial number. If one wants linear radial Regge behavior, the soft wall or a more dynamical confining background is more natural.

Mistake 4: ignoring boundary conditions

IR boundary conditions are part of the model. Changing Dirichlet to Neumann conditions can change the spectrum and couplings. These choices should be justified by symmetry, variational principles, or phenomenological calibration.

Mistake 5: treating $m_q$ and $\sigma$ too naively

In simple hard-wall models, the quark mass and condensate coefficients are often independent inputs. That is not the same as a dynamical derivation of chiral symmetry breaking.

Mistake 6: thinking the dilaton factor always affects strings the same way it affects probe fields

The soft-wall factor $e^{-\Phi}$ in a five-dimensional meson action modifies probe-field wave equations. A Wilson loop probes the string-frame metric and fundamental string action. These are related only in a fully specified string background.

Exercises

Exercise 1: Hard-wall vector modes

Consider a transverse five-dimensional gauge field in AdS$_5$ on the interval $0<z\le z_m$. Starting from

$$\partial_z\left(\frac1z\partial_z v_n\right) + \frac{m_n^2}{z}v_n=0,$$

show that the regular solution is $v_n(z)=C_n zJ_1(m_nz)$. Then impose the Neumann condition $\partial_zv_n(z_m)=0$ and derive the quantization condition.

Solution

Multiply the equation by $z$:

$$v_n''-\frac1zv_n'+m_n^2v_n=0.$$

Let $v_n(z)=z f(m_nz)$ and define $u=m_nz$. Then

$$u^2 f''+u f'+(u^2-1)f=0,$$

which is Bessel’s equation of order one. The solution regular at $z=0$ is

$$f(u)=J_1(u),$$

so

$$v_n(z)=C_n zJ_1(m_nz).$$

The Neumann condition gives

$$0=\partial_z\left[zJ_1(m_nz)\right]_{z=z_m} =m_nz_mJ_0(m_nz_m).$$

Therefore

$$J_0(m_nz_m)=0, \qquad m_n=\frac{\gamma_{0,n}}{z_m},$$

where $\gamma_{0,n}$ is the $n$th zero of $J_0$.

Exercise 2: The soft-wall Schrödinger potential

For the soft-wall vector equation

$$\partial_z\left(e^{-B}\partial_z v_n\right) +m_n^2e^{-B}v_n=0, \qquad B(z)=\kappa^2z^2+\log z,$$

set $v_n=e^{B/2}\psi_n$. Show that

$$-\psi_n''+ \left(\kappa^4z^2+\frac{3}{4z^2}\right)\psi_n =m_n^2\psi_n.$$

Solution

Start by expanding the equation:

$$v_n''-B'v_n'+m_n^2v_n=0.$$

With $v_n=e^{B/2}\psi_n$,

$$v_n'=e^{B/2}\left(\psi_n'+\frac{B'}2\psi_n\right),$$

and

$$v_n''=e^{B/2}\left[ \psi_n''+B'\psi_n' +\left(\frac{B''}{2}+\frac{B'^2}{4}\right)\psi_n \right].$$

Substitution gives

$$\psi_n''+ \left(\frac{B''}{2}-\frac{B'^2}{4}\right)\psi_n +m_n^2\psi_n=0.$$

Therefore

$$-\psi_n''+ \left(\frac{B'^2}{4}-\frac{B''}{2}\right)\psi_n =m_n^2\psi_n.$$

For $B=\kappa^2z^2+\log z$,

$$B'=2\kappa^2z+\frac1z, \qquad B''=2\kappa^2-\frac1{z^2}.$$

Thus

$$\frac{B'^2}{4}-\frac{B''}{2} = \kappa^4z^2+\kappa^2+\frac{1}{4z^2} -\kappa^2+\frac{1}{2z^2} = \kappa^4z^2+\frac{3}{4z^2}.$$

Exercise 3: Source and condensate for the chiral scalar

The bifundamental scalar $X$ is dual to $\bar q_Rq_L$ with $\Delta=3$ in a four-dimensional boundary theory. Use the scalar mass-dimension relation to find $m_X^2L^2$, and write the near-boundary expansion of $X$.

Solution

For a scalar in AdS$_5$/CFT$_4$,

$$m^2L^2=\Delta(\Delta-4).$$

With $\Delta=3$,

$$m_X^2L^2=3(3-4)=-3.$$

The two independent near-boundary powers are

$$z^{4-\Delta}=z, \qquad z^\Delta=z^3.$$

Therefore the standard hard-wall ansatz is

$$X(z) \sim \frac12(m_qz+\sigma z^3)\mathbf 1_{N_f},$$

where $m_q$ is the quark-mass source and $\sigma$ is proportional to the condensate $\langle\bar q q\rangle$.

Exercise 4: Why the hard wall misses linear radial Regge behavior

Using the large-$n$ asymptotic behavior of Bessel zeros,

$$\gamma_{0,n}\sim \left(n-\frac14\right)\pi,$$

show that the hard-wall vector spectrum has $m_n^2\sim n^2$. Compare this with the soft-wall vector spectrum.

Solution

The hard-wall vector masses are

$$m_n=\frac{\gamma_{0,n}}{z_m}.$$

For large $n$,

$$m_n\sim \frac{\pi n}{z_m},$$

so

$$m_n^2\sim \frac{\pi^2n^2}{z_m^2}.$$

The soft-wall vector spectrum is instead

$$m_n^2=4\kappa^2(n+1),$$

so

$$m_n^2\sim n.$$

This is why the soft wall is better suited to modeling linear radial Regge trajectories.

Exercise 5: Which confinement diagnostic is being tested?

Classify each statement as testing a mass gap, Wilson-loop confinement, chiral symmetry breaking, or large-$N_c$ resonance behavior.

1. The two-point function has isolated poles at $q^2=-m_n^2$.
2. The scalar $X$ has a nonzero $z^3$ coefficient when $m_q\to0$.
3. A large rectangular Wilson loop gives $V(R)\sim \sigma R$.
4. Meson widths vanish as $N_c\to\infty$.

Solution

1. Isolated poles indicate a discrete spectrum and therefore a mass gap when the lowest pole is at nonzero mass.
2. A nonzero condensate coefficient in the chiral limit indicates spontaneous chiral symmetry breaking, assuming the coefficient is dynamically determined rather than merely inserted.
3. A linear heavy-quark potential tests Wilson-loop confinement.
4. Vanishing widths are the large-$N_c$ resonance property. In holography this corresponds to tree-level classical bulk fields and suppressed bulk interactions.

Further reading

• J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, QCD and a Holographic Model of Hadrons. The classic hard-wall AdS/QCD model for low-energy meson physics.
• A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear Confinement and AdS/QCD. The original soft-wall model emphasizing linear Regge behavior.
• J. Polchinski and M. J. Strassler, Hard Scattering and Gauge/String Duality. An influential use of an IR cutoff in gauge/string duality for confining physics.
• J. Erdmenger, N. Evans, I. Kirsch, and E. Threlfall, Mesons in Gauge/Gravity Duals. A useful broader review of mesons in holographic models.
• U. Gursoy, E. Kiritsis, F. Nitti, and collaborators on improved holographic QCD. These models go beyond fixed AdS backgrounds by using dynamical Einstein-dilaton geometries fitted to QCD-like thermodynamics and spectra.