# Comparison between holographic deformed AdS and soft wall models for
fermions

Ayrton da Cruz Pereira do Nascimento<EMAIL_ADDRESS>Henrique Boschi-
Filho<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro, 21.941-909 - Rio de Janeiro - RJ - Brazil

###### Abstract

We compare the holographic dressed soft wall and the exponentially deformed
AdS models for spin 1/2 fermions. We present the dressed soft wall model and
its analytical solutions for the left and right modes, and the corresponding
spectra, also including modifications considering hyperfine spin-spin and
meson cloud interactions, as well as anomalous dimensions. Then, we discuss
the deformed AdS model for spin 1/2 fermions and present their effective
Schrödinger equations for the left and right modes, for which only numerical
solutions are available. Then, we consider a polynomial expansion of the
effective potential of the deformed AdS model and show that in the quadratic
approximation it leads to exact analytical solutions comparable with the
dressed soft wall model and obtain the corresponding spectra for left and
right modes. We show a numerical comparison of the mass spectra of spin 1/2
baryons for the dressed soft wall and the deformed AdS models. We present a
detailed relation between the quadratic approximation of the deformed AdS and
the dressed soft wall models for their spectra, wave functions and comments on
the deep inelastic scattering on both models. We find that these two models
are not equivalent even in the quadratic approximation, but it is possible to
relate their left and right modes for particular choices of their parameters.

###### Contents

  1. I Introduction
  2. II Dressed Soft Wall Model
    1. II.1 Anomalous Dressed Soft Wall Model
  3. III Deformed AdS Model
    1. III.1 Quadratic approximation
  4. IV Comment on the Signs of the Dilaton for Fermions
  5. V Numerical comparison of the models
  6. VI Analytic comparison of the models
    1. VI.1 Spectra
    2. VI.2 Wave Functions
    3. VI.3 Comments on the DIS in both models
  7. VII Conclusions
  8. A The isotonic oscillator
  9. B Case $(ii)$: $m_{5}=-\tilde{m}_{5}\mp 1$
  10. C Comment on relations with top-down models

##  I Introduction

The AdS/CFT correspondence, in its most well known form, relates a five
dimensional gravitational theory in anti de-Sitter space with a conformal non-
gravitational field theory in four dimensions Maldacena:1997re ;
Aharony:1999ti , in accordance with the holographic principle tHooft:1993dmi ;
Susskind:1994vu . This relation is useful to describe difficult problems in
non-perturbative models, since a weakly coupled 5-d gravitational theory is
dual to a strongly coupled 4-d field theory. This has applications in many
areas of physics and in particular for low energy QCD. This is made possible
breaking the conformal symmetry of the original AdS/CFT proposal in different
ways, which became known as the AdS/QCD models.

This was first realized in the holographic hard wall model, which constitutes
a slice of finite size in the holographic coordinate in AdS space introduced
to describe the hard scattering Polchinski:2001tt  and spectra Boschi-
Filho:2002xih ; Boschi-Filho:2002wdj  of scalar glueballs, and deep inelastic
scattering (DIS) of scalars and fermions Polchinski:2002jw . Later, this model
was extended to include the spectra of mesons Erlich:2005qh ;
deTeramond:2005su , higher spin glueballs deTeramond:2005su ; Boschi-
Filho:2005xct  and baryons deTeramond:2005su . Besides these achievements, one
weak aspect of the hard wall model is that it leads to non-linear (parabolic)
Regge trajectories, in contrast with the observed linear hadronic spectra.

This point was overcome by the holographic soft wall model, which introduces
an exponential factor in the AdS bulk action, leading to discrete spectra for
mesons with linear Regge trajectories Karch:2006pv ; Karch:2010eg ;
Colangelo:2008us . Similar Regge trajectories are obtained from this model for
glueballs Colangelo:2007pt , although in this case the spectra is not good in
comparison with lattice data. This discrepancy motivated the proposal of
modifications in the original holographic soft wall model, as discussed for
instance in Gursoy:2007cb ; Gursoy:2007er ; Li:2013oda . In contrast with the
hard wall, the original soft wall model does not give a discrete spectrum for
fermions due to the bilinear form of the Dirac action. In this case the
exponential factor is simply absorbed in a normalization of the fermionic
fields. The first soft wall description of DIS was given by
BallonBayona:2007qr  for mesons and where a hybrid model with an additional
hard wall was used to discretize the fermionic spectra.

An extension of the soft wall model which gives discrete spectra for spin 1/2
fermions was proposed in Abidin:2009hr , where there is a supplementary
interaction with the background field by using a scalar potential in the mass
term. Since this term depends only on the holographic coordinate, this model
is equivalent to a soft wall with a curvature dependent mass in the five
dimensional AdS bulk which can also be understood as a dressed mass. This
model has been successfully used to study mesons and baryons Branz:2010ub ;
Gutsche:2011vb , and to obtain the DIS proton structure functions Braga:2011wa
.

Recently, it appeared in the literature an AdS/QCD model with an exponential
deformation in the AdS metric, instead of the case of the soft wall model
where the exponential factor is included in the action. This exponentially
deformed AdS model provides good spectra for hadrons of various spins
FolcoCapossoli:2019imm , the DIS proton structure functions
FolcoCapossoli:2020pks , the pion MartinContreras:2021yfz , proton and neutron
Contreras:2021epz  form factors, and in the Brownian motion Caldeira:2020sot ;
Caldeira:2020rir ; Caldeira:2021izy . This model was inspired by the obtention
of a confining quark-antiquark potential from the soft wall model
Andreev:2006ct ; Bruni:2018dqm . It is important to notice that the deformed
AdS model when applied to mesons gives exactly the same spectra as the
original soft wall model. However, for any other hadrons the spectra are
different.

Here, we consider two holographic AdS/QCD models, namely the soft wall model
with a curvature dependent or dressed mass Abidin:2009hr ; Branz:2010ub ;
Gutsche:2011vb , which for simplicity we are just calling it dressed soft wall
and the exponentially deformed AdS/QCD model FolcoCapossoli:2019imm . These
models have been used recently to study proton structure functions in DIS
Braga:2011wa ; FolcoCapossoli:2020pks , providing similar results. One
remarkable difference between these two models is that the dressed soft wall
is analytically solvable for fermions while the in deformed AdS the
corresponding equations are only numerically tractable. We show a numerical
comparison of the spectra of these models. In the case of the dressed soft
wall model we also include hyperfine spin-spin and meson cloud interactions
Gutsche:2011vb , as well as anomalous dimensions affecting the fermionic
spectra, following the ideas of Refs. Boschi-Filho:2012ijd ;
FolcoCapossoli:2016uns ; Rodrigues:2016kez . In both models, the spectra come
from the solution of pairs of effective Schrödinger equations, usually
labelled as left and right modes. In the case of the dressed soft wall, these
equations reduce to a harmonic oscillators plus centrifugal barriers in
opposition with the exponentially deformed AdS model with more involved
effective potentials. Then, we consider a quadratic approximation for the
effective potentials of the deformed AdS model find their analytical solutions
and compare them with the ones coming from the dressed soft wall model. We
show that the two models are not equivalent even in the quadratic
approximation, but it is possible to relate the left and right modes of both
models for particular choices of their parameters.

This work is organized as follows: In Section II, we briefly review the
dressed soft wall model for spin 1/2 fermions, obtaining their effective
Schrödinger like equations of motion, the corresponding analytical solutions,
spectra, and a modification of the fermionic spectra with the inclusion of
corrections from hyperfine spin-spin and meson cloud interactions. In
Subsection A, we present a fermionic spectrum modified by the inclusion of
anomalous dimensions. In Section III, we discuss the exponentially deformed
AdS model for spin 1/2 fermions and write their effective Schrödinger like
equations of motion for the left and right modes, consider the expansion of
the exponential deformation in the effective potentials as power series and
restrict them to a quadratic approximation, which we solve analytically. Then,
in Section IV, we present a brief discussion on the sign of the dilaton in
these two models. In Section V, we compare spin 1/2 baryonic spectra in the
dressed soft wall model (including hyperfine spin-spin and meson cloud
interactions and anomalous dimensions) and the deformed AdS model. In section
VI, we perform an analytical comparison of the dressed soft wall effective
Schrödinger like equations and the corresponding ones for the deformed AdS in
the quadratic regime and discuss the possible relations between these models
in terms of spectra, wave functions and DIS interaction actions. Finally, in
section VII, we present our conclusions. We also include three Appendices
where we give some additional details of the material presented in the main
text and discuss briefly the relation of the analytic soft walls with top-down
D$p$-brane models Kirsch:2006he ; Abt:2019tas ; Nakas:2020hyo .

##  II Dressed Soft Wall Model

The original soft wall model is defined by the action Karch:2006pv ;
Colangelo:2008us

$S=\int dz\,d^{4}{x}\sqrt{-g}\,e^{-\Phi(z)}\,\mathcal{L},$ (1)

where $\Phi(z)=\kappa^{2}\,z^{2}$ is the dilaton, $\mathcal{L}$ is the field
lagrangian and $g$ is the determinant of the metric $g_{mn}$ of the pure
$AdS_{5}$ space:

$ds^{2}=g_{mn}\,dx^{m}dx^{n}=\frac{\Omega^{2}}{z^{2}}(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}),$
(2)

with $\Omega$ being the AdS radius, $z$ the holographic coordinate, and
$\eta_{\mu\nu}$ the four dimensional flat Minkowski metric with signature
$(-,+,+,+)$. The dilaton $\Phi(z)$ plays the role of a soft wall bringing in a
discrete spectra, at least for mesons, and $\kappa$ which has dimension of
mass is the IR scale of the model.

A modified version of this model is required if we want to describe fermions
with discrete mass spectra. This is so because in an standard fermionic action
the exponential factor $e^{-\Phi(z)}$ would drop out from the equations of
motion, thus not affecting their spectra, due to the bilinear form of the
Dirac Lagrangian. This issue can be overcame introducing a potential ${\cal
V}(z)$ in the mass term, which can be interpreted as a dressed mass in the
action Abidin:2009hr , such that the fermionic spectra becomes discrete. This
is the dressed soft wall model. The fermionic action for spin 1/2 in this case
is written as (following the notation of Ref. Braga:2011wa ):

$S=\int
dz\,d^{4}x\,\sqrt{-g}\,e^{-\Phi(z)}\,\left[\frac{i}{2}\bar{\Psi}\,e^{m}_{a}\,\Gamma^{a}\,\mathcal{D}_{m}\Psi-\frac{i}{2}(\mathcal{D}_{m}\,\Psi)^{\dagger}\,\Gamma^{0}\,e^{m}_{a}\,\Gamma^{a}\,\Psi-\bar{\Psi}\left(\mu_{5}+{\cal
V}(z)\right)\,\Psi\right],$ (3)

with $\mu_{5}$ a $5d$ constant fermion bulk mass and ${\cal
V}(z)=\kappa^{2}\,z^{2}/\Omega$ was chosen to lead to exact solutions for the
fermionic spectra. The dressed mass of this model is $\mu_{5}+{\cal V}(z)$.
The covariant derivative is given by

$\mathcal{D}_{m}\equiv\partial_{m}+\frac{1}{2}\omega_{m}^{bc}\,\Sigma_{bc},$
(4)

where $\Sigma_{bc}=\frac{1}{4}[\Gamma_{b},\Gamma_{c}]$, with Dirac matrices
$\Gamma^{a}=(\gamma^{\mu},-i\gamma^{5})$. The factor $e^{a}_{n}$ is the
vielbein which couples the fermion to the $AdS_{5}$ space,

$e^{m}_{a}=\frac{z}{\Omega}\,\delta^{m}_{a},$ (5)

with $m=0,1,2,3,5$, and $\omega_{m}^{bc}$ is the spin connection,

$\omega^{a\,b}_{m}=-\frac{\Omega}{z}\left(\delta^{a}_{z}\delta^{b}_{m}-\delta^{b}_{z}\delta^{a}_{m}\right).$
(6)

After a rescaling of the fermionic field

$\Psi(x^{\mu},z)=e^{\kappa^{2}\,z^{2}/2}\psi(x^{\mu},z),$ (7)

the action reads

$S=\int
dz\,d^{4}x\,\sqrt{-g}\,\left[\frac{i}{2}\bar{\psi}\,e^{m}_{a}\,\Gamma^{a}\,\mathcal{D}_{m}\psi-\frac{i}{2}(\mathcal{D}_{m}\,\psi)^{\dagger}\,\Gamma^{0}\,e^{m}_{a}\,\Gamma^{a}\,\psi-\bar{\psi}\left(\mu_{5}+\frac{\kappa^{2}z^{2}}{\Omega}\right)\,\psi\right].$
(8)

Now, considering a field decomposition into left and right chiral components

$\psi(x^{\mu},z)=\psi_{{}_{L}}(x^{\mu},z)+\psi_{{}_{R}}(x^{\mu},z),$ (9)

one can write each chiral mode in terms of the four-dimensional Dirac spinors
$u_{s}(P)$ as

$\psi_{\frac{L}{R}}(x^{\mu},z)=e^{iP\cdot
x}\frac{1}{2}(1\mp\gamma^{5})u_{s}(P)\frac{z^{2}}{\Omega^{2}}{f}_{\frac{L}{R}}(z),$
(10)

where $P^{\mu}$ is the four-dimensional momenta of the hadron. The equations
of motion can be written as Abidin:2009hr ; Branz:2010ub ; Gutsche:2011vb ;
Braga:2011wa

$\left[-\partial_{z}^{2}+\kappa^{4}\,z^{2}+2\,\kappa^{2}\left(m_{5}\mp\frac{1}{2}\right)+\frac{m_{5}(m_{5}\pm
1)}{z^{2}}\right]{f}^{n}_{\frac{L}{R}}(z)=M_{n}^{2}\,{f}^{n}_{\frac{L}{R}}(z),$
(11)

where

$m_{5}=\mu_{5}\,\Omega$ (12)

is a dimensionless mass parameter of the five dimensional theory and $M_{n}$
are the hadron masses. The corresponding spectrum reads (see Appendix A)

$M_{n\frac{L}{R}}^{2}=2\,\kappa^{2}(m_{5}\mp\frac{1}{2})+2(2n+1)\kappa^{2}+\sqrt{\kappa^{4}[1+4m_{5}(m_{5}\pm
1)]};\qquad(n=0,1,2,\dots),$ (13)

so that

$M_{n\frac{L}{R}}^{2}=2\kappa^{2}\left(m_{5}\mp\frac{1}{2}+2n+1+\left|m_{5}\pm\frac{1}{2}\right|\right);\qquad(n=0,1,2,\dots).$
(14)

If $m_{5}\geq 1/2$ the fractions with $\mp$ and $\pm$ cancel, leading exactly
to the same spectrum for the right and left modes:

$M_{n}^{2}=4\,\kappa^{2}\left(n+m_{5}+\frac{1}{2}\right);\qquad\qquad(n=0,1,2,\dots).$
(15)

On the other side, if $m_{5}\leq-1/2$, the terms with $m_{5}$ cancel and the
spectrum reads

$M_{n}^{2}=4\,\kappa^{2}\left(n+1\right);\qquad\qquad(n=0,1,2,\dots).$ (16)

If $m_{5}$ belongs to the interval $(-1/2,1/2)$, the spectrum for the left and
right modes are different and given by Eq. (14).

In all of these cases, the corresponding Regge trajectories are linear as
expected for the soft wall, and the normalized solutions are given by

$\displaystyle{f}^{n}_{{}_{L/R}}(z)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+|m_{5}\pm
1/2|+1)}}\,|\kappa|^{|m_{5}\pm 1/2|+1}\,z^{|m_{5}\pm
1/2|+1/2}\,e^{-\kappa^{2}z^{2}/2}\,L^{|m_{5}\pm 1/2|}_{n}(\kappa^{2}z^{2}),$
(17)

where $L^{p}_{n}(\kappa^{2}\,z^{2})$ are the associated Laguerre polynomials
(see Appendix A).

In particular, for $m_{5}\geq 1/2$, the normalized solutions
${f}^{n}_{L/R}(z)$ are given by

$\displaystyle{f}^{n}_{{}_{L}}(z)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+m_{5}+3/2)}}\,|\kappa|^{m_{5}+3/2}\,z^{m_{5}+1}\,e^{-\kappa^{2}z^{2}/2}\,L^{m_{5}+1/2}_{n}(\kappa^{2}z^{2}),$
(18a) $\displaystyle{f}^{n}_{{}_{R}}(z)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+m_{5}+1/2)}}\,|\kappa|^{m_{5}+1/2}\,z^{m_{5}}\,e^{-\kappa^{2}z^{2}/2}\,L^{m_{5}-1/2}_{n}(\kappa^{2}z^{2}),$
(18b)

recovering the results of Ref. Braga:2011wa .

In order to find a baryonic spectra from the above discussion, one can start
from the standard relation of the AdS/CFT dictionary for spin 1/2 fermions,
$|m_{5}|=\Delta-2$, and consider that each fermion in $d=4$ spacetime
dimensions has conformal dimension 3/2, so a baryon with three constituent
quarks imply $\Delta=9/2$, so that $m_{5}=\pm 5/2$. Taking for simplicity
$m_{5}=5/2$, then Eq. (15) reads

$M_{n}^{2}=4\,\kappa^{2}\left(n+3\right);\qquad\qquad(n=0,1,2,\dots).$ (19)

In Ref. Gutsche:2011vb , the authors considered the mass parameter for spin
1/2 fermions in $d=4$ spacetime dimensions as $m_{5}=\ell+3/2$, where
$\ell=0,1$. Then, Eq. (15) gets

$M_{n,\ell}^{2}=4\,\kappa^{2}\left(n+\ell+2\right);\qquad\qquad(n=0,1,2,\dots).$
(20)

and the wave functions can be rewritten as

$\displaystyle{f}^{n}_{{}_{L}}(z)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+\ell+3)}}\,\kappa^{\ell+3}\,z^{\ell+5/2}\,e^{-\kappa^{2}z^{2}/2}\,L^{\ell+2}_{n}(\kappa^{2}z^{2}),$
(21a) $\displaystyle{f}^{n}_{{}_{R}}(z)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+\ell+2)}}\,\kappa^{\ell+2}\,z^{\ell+3/2}\,e^{-\kappa^{2}z^{2}/2}\,L^{\ell+1}_{n}(\kappa^{2}z^{2}).$
(21b)

In this case, the spectra and eigenfunctions are degenerate since
$n=0,1,2,...$ and $\ell=0,1$. So, using some phenomenological arguments
related to hyperfine spin-spin and meson cloud interactions, they find a
formula for baryonic spectra Gutsche:2011vb

$M_{{}_{B}}^{2}=4\,\kappa^{2}\left[n+\ell+2+\alpha_{{}_{\rm
HF}}(S-1)-\beta_{{}_{\rm MC}}\right];\qquad\qquad(n=0,1,2,\dots).$ (22)

with $\kappa=$ 500 MeV, $\alpha_{{}_{\rm HF}}=0.636$ and $\beta_{{}_{\rm
MC}}=0.800$. With these parameters they were able the obtain, for instance,
the spectra of $N(1/2^{+})$ baryons with $S=1/2$ and $\ell=0$, which are shown
in Table 1.

###  II.1 Anomalous Dressed Soft Wall Model

Alternatively, it is also possible to modify the spectrum given, for instance,
by Eqs. (15) and (19), taking into account anomalous dimensions in the
conformal dimension of the baryonic operators. This follows the idea of
introducing anomalous dimensions in the soft wall model for glueball operators
Boschi-Filho:2012ijd ; FolcoCapossoli:2016uns ; Rodrigues:2016kez , and was
also used in the numerical solution of the deformed AdS model for baryons
FolcoCapossoli:2020pks .

Here, we use the idea of introducing an anomalous dimension in the analytical
baryonic spectrum, given by Eq. (19). Then, taking $|m_{5}|=\Delta_{\rm
canonical}-2+\gamma$, one has

$M_{{n}_{A}}^{2}=4\,\kappa^{2}\left(n+3+\gamma\right);\qquad\qquad(n=0,1,2,\dots),$
(23)

where the anomalous dimension $\gamma$ is usually a function of the spin. For
instance, if one has $\gamma=-2$, one finds

$M_{{n}_{A}}^{2}=4\,\kappa^{2}\left(n+1\right);\qquad\qquad(n=0,1,2,\dots).$
(24)

Clearly, comparing eqs. (22), (23), and (24), one sees that the consideration
of hyperfine spin-spin and meson cloud interactions in Ref. Gutsche:2011vb  is
equivalent to including anomalous dimensions in the conformal dimension of
baryonic operators. The masses calculated according to the above formula are
also presented in Table 1.

##  III Deformed AdS Model

In this section we discuss the deformed AdS model. The 5-dimensional action is
given by FolcoCapossoli:2019imm

$\tilde{S}=\int d\tilde{z}\,d^{4}x\,\sqrt{-\tilde{g}}\,\tilde{\mathcal{L}},$
(25)

where $\tilde{\mathcal{L}}$ is the lagrangian density and $\tilde{g}$ is the
determinant of the metric of the exponentially deformed $AdS_{5}$ space

$d\tilde{s}^{2}=\tilde{g}_{mn}\,d\tilde{x}^{m}d\tilde{x}^{n}=e^{2\,A(\tilde{z})}\,(d\tilde{z}^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}),$
(26)

$\tilde{z}$ being the holographic coordinate and

$A(\tilde{z})=-\log\left(\frac{\tilde{z}}{\tilde{\Omega}}\right)+\frac{1}{2}\,k\,\tilde{z}^{2}.$
(27)

The constant $k$ has dimension of mass squared. Note that in the asymptotic
limit $k\rightarrow 0$, the pure AdS space with radius $\tilde{\Omega}$ is
obtained.

Now, we address the question of how to describe fermionic fields in this
deformed background model. To do so, we follow the steps in
FolcoCapossoli:2020pks , starting with the fermionic action given by

$\tilde{S}=\int
d\tilde{z}\,d^{4}x\,\sqrt{\tilde{g}}\,\tilde{\bar{\Psi}}(\tilde{\not{D}}-\tilde{\mu}_{5})\tilde{\Psi},$
(28)

where $\tilde{\mu}_{5}$ is a constant fermionic bulk mass and the operator
$\tilde{\not{D}}$ is defined as:

$\tilde{\not{D}}\equiv\tilde{g}^{mn}\,\tilde{e}^{a}_{n}\,\Gamma_{a}\left(\partial_{m}+\frac{1}{2}\tilde{\omega}_{m}^{bc}\,\Sigma_{bc}\right).$
(29)

Note that the Dirac matrices $\Gamma_{a}$ and $\Sigma_{bc}$ were defined in
the previous section, and here the vielbein $\tilde{e}^{a}_{n}$ is given by

$\tilde{e}^{a}_{m}=e^{A(\tilde{z})}\,\delta^{a}_{m}.$ (30)

The spin connection $\tilde{\omega}_{m}^{bc}$ is

$\tilde{\omega}^{a\,b}_{m}=\tilde{e}^{a}_{n}\,\partial_{m}\tilde{e}^{n\,b}+\tilde{e}^{a}_{n}\,\tilde{e}^{p\,b}\,\tilde{\Gamma}^{n}_{p\,m},$
(31)

with $m=0,1,2,3,5$, and $\tilde{\Gamma}^{n}_{p\,m}$ represent the Christoffel
symbols which can be written as

$\tilde{\Gamma}^{p}_{m\,n}=\frac{1}{2}\,\tilde{g}^{p\,q}(\partial_{n}\,\tilde{g}_{m\,q}+\partial_{m}\,\tilde{g}_{n\,q}-\partial_{q}\,\tilde{g}_{m\,n}).$
(32)

From Eq. (28), one can derive the following equation of motion
FolcoCapossoli:2020pks

$(e^{-A(\tilde{z})}\,\gamma^{5}\partial_{\tilde{z}}+e^{-A(\tilde{z})}\,\gamma^{\mu}\partial_{\mu}+2\,A^{\prime}(\tilde{z})\gamma^{5}-\tilde{\mu}_{5})\,\tilde{\Psi}=0,$
(33)

and prime denote derivative with respect to $\tilde{z}$. As before, the
fermionic fields are written in terms of left and right chiral components

$\tilde{\Psi}(x^{\mu},\tilde{z})=\tilde{\Psi}_{{}_{L}}(x^{\mu},\tilde{z})+\tilde{\Psi}_{{}_{R}}(x^{\mu},\tilde{z}),$
(34)

which can be written as

$\tilde{\Psi}_{\frac{L}{R}}(x^{\mu},\tilde{z})=e^{iP\cdot
x}\frac{1}{2}(1\mp\gamma^{5})u_{s}(P)\frac{\tilde{z}^{2}}{{\tilde{\Omega}}^{2}}\tilde{f}_{\frac{L}{R}}(\tilde{z})\,e^{-2\,A(\tilde{z})}.$
(35)

Then, one gets an effective Schrödinger equation for each chiral mode
$\tilde{f}_{\frac{L}{R}}(\tilde{z})$ FolcoCapossoli:2020pks

$[-\partial^{2}_{\tilde{z}}+V_{\frac{L}{R}}(\tilde{z})]\,\tilde{f}_{\frac{L}{R}}^{n}(\tilde{z})=\tilde{M}_{n}^{2}\,\tilde{f}^{n}_{\frac{L}{R}}(\tilde{z}),$
(36)

where $\tilde{M}_{n}$ is the $n$-th fermionic mass in four dimensions in the
exponentially deformed AdS model and $V_{\frac{L}{R}}(\tilde{z})$ are the
chiral potentials

$V_{\frac{L}{R}}(\tilde{z})=\tilde{\mu}_{5}^{2}\,e^{2\,A(\tilde{z})}\mp\,e^{A(\tilde{z})}\,\tilde{\mu}_{5}\,A^{\prime}(\tilde{z}).$
(37)

With these potentials, the effective Schrödinger equations, Eq. (36), in
general, cannot be solved analytically and were solved numerically in
FolcoCapossoli:2019imm ; FolcoCapossoli:2020pks . In the next subsection, we
implement a perturbative approach for these potentials, leading to approximate
analytical solutions for the chiral modes
$\tilde{f}^{n}_{\frac{L}{R}}(\tilde{z})$.

###  III.1 Quadratic approximation

In order to perform an analytical comparison of the two models, here we make a
quadratic approximation on the exponentially deformed AdS model. In the
deformed AdS model, we have effective Schrödinger equations for each chiral
mode $\tilde{f}^{n}_{R/L}(\tilde{z})$ with chiral potentials, Eq. (37). These
potentials depend on exponentials of the warp factor $A(\tilde{z})$, given by
equation (27), which can be written as

$\displaystyle e^{A(\tilde{z})}$
$\displaystyle=\frac{\tilde{\Omega}}{\tilde{z}}\,e^{k\,\tilde{z}^{2}/2}.$ (38)

Introducing $\xi=k\,\tilde{z}^{2}$, one can expand the exponentials to second
order in this new variable, to obtain

$\displaystyle e^{A(\tilde{z})}$
$\displaystyle=\frac{\tilde{\Omega}}{\tilde{z}}\,e^{\xi/2}=\frac{\tilde{\Omega}}{\tilde{z}}\left(1+\frac{\xi}{2}+\frac{1}{8}\xi^{2}+{\cal
O}(\xi^{3})\right)\approx\frac{\tilde{\Omega}}{z}+\frac{k\,\tilde{z}\,\tilde{\Omega}}{2}+\frac{1}{8}k^{2}\,\tilde{z}^{3}\,\tilde{\Omega},$
(39a) $\displaystyle e^{2\,A(\tilde{z})}$
$\displaystyle=\frac{\tilde{\Omega}^{2}}{\tilde{z}^{2}}\,e^{\xi}=\frac{\tilde{\Omega}^{2}}{\tilde{z}^{2}}\,\left(1+\xi+\frac{1}{2}\,\xi^{2}+{\cal
O}(\xi^{3})\right)\approx\frac{\tilde{\Omega}^{2}}{\tilde{z}^{2}}+\tilde{\Omega}^{2}\,k+\frac{\tilde{\Omega}^{2}}{2}\,k^{2}\,\tilde{z}^{2}.$
(39b)

We also have

$A^{\prime}(\tilde{z})=\frac{d}{d\,\tilde{z}}\left(-\log\left(\frac{\tilde{z}}{\tilde{\Omega}}\right)+\frac{k\,\tilde{z}^{2}}{2}\right)=-\frac{1}{\tilde{z}}+k\,\tilde{z}.$
(40)

Using (39a) alongside with (40), and keeping terms up to $\tilde{z}^{2}$, we
get

$e^{A(\tilde{z})}\,A^{\prime}(\tilde{z})=\left(\frac{\tilde{\Omega}}{\tilde{z}}+\frac{k\,\tilde{z}\,\tilde{\Omega}}{2}+\frac{1}{8}k^{2}\,\tilde{z}^{3}\,\tilde{\Omega}\right)\left(-\frac{1}{\tilde{z}}+k\,\tilde{z}\right)\approx-\frac{\tilde{\Omega}}{\tilde{z}^{2}}+\frac{k\,\tilde{\Omega}}{2}+\frac{3k^{2}\,\tilde{z}^{2}\,\tilde{\Omega}}{8}.$
(41)

Substituting (39b) and (41) in (37), we obtain the effective potential

$V_{\frac{L}{R}}(\tilde{z})=\tilde{\mu}_{5}^{2}\tilde{\Omega}^{2}\left(\frac{1}{\tilde{z}^{2}}+k+\frac{1}{2}\,k^{2}\,\tilde{z}^{2}\right)\mp\tilde{\mu}_{5}\tilde{\Omega}\left(-\frac{1}{\tilde{z}^{2}}+\frac{k}{2}+\frac{3}{8}\,k^{2}\,\tilde{z}^{2}\right),$
(42)

or equivalently

$V_{\frac{L}{R}}(\tilde{z})=\frac{k_{1\frac{L}{R}}}{\tilde{z}^{2}}+k_{2\frac{L}{R}}+{k_{3\frac{L}{R}}}\,\tilde{z}^{2},$
(43)

with

$\displaystyle k_{1\frac{L}{R}}$
$\displaystyle=\tilde{m}_{5}^{2}\pm\tilde{m}_{5},$ (44a) $\displaystyle
k_{2\frac{L}{R}}$
$\displaystyle=\left(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2}\right)\,k,$
(44b) $\displaystyle k_{3\frac{L}{R}}$
$\displaystyle=\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)\,k^{2},$
(44c)

where we have introduced the dimensionless parameter

$\tilde{m}_{5}=\tilde{\mu}_{5}\,\tilde{\Omega}.$ (45)

Then, the effective Schrödinger equations (36) for each chiral mode in the
quadratic approximation read

$\left[-\partial_{\tilde{z}}^{2}+\frac{k_{1\frac{L}{R}}}{\tilde{z}^{2}}+k_{2\frac{L}{R}}+k_{3\frac{L}{R}}\,\tilde{z}^{2}\right]\,\tilde{f}_{\frac{L}{R}}^{n}(\tilde{z})=M_{n}^{2}\,\tilde{f}_{\frac{L}{R}}^{n}(\tilde{z}).$
(46)

We thus have a Schrödinger equation with a harmonic term plus a
centrifugal/centripetal barrier term falling off as $1/\tilde{z}^{2}$. Such
equations have analytical solutions which can be read off as (see Appendix A)

$\tilde{f}_{\frac{L}{R}}^{n}(\tilde{z})=\sqrt{\frac{2n!\,\left(k_{3\frac{L}{R}}\right)^{\frac{1}{2}\left(1+\phi_{\frac{L}{R}}\right)}}{\Gamma(n+\phi_{\frac{L}{R}}+1)}}\,\tilde{z}^{\left(\frac{1}{2}+\phi_{\frac{L}{R}}\right)}\,\exp{\left(-\frac{1}{2}\,\sqrt{k_{3\frac{L}{R}}}\,\tilde{z}^{2}\right)}\,L^{\left(\phi_{\frac{L}{R}}\right)}_{n}\left(\sqrt{k_{3\frac{L}{R}}}\,\tilde{z}^{2}\right),$
(47)

with

$\phi_{\frac{L}{R}}=\frac{1}{2}\sqrt{1+4\,k_{1\frac{L}{R}}}.$ (48)

The spectrum is given by

$\displaystyle M_{n\frac{L}{R}}^{2}$
$\displaystyle=k_{2\,\frac{L}{R}}+2\left[(2\,n+1)\,\sqrt{k_{3\,\frac{L}{R}}}\right]+\sqrt{k_{3\,\frac{L}{R}}\,(1+4\,k_{1\,\frac{L}{R}})}$
$\displaystyle=(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2})\,k+2\,|k|\,\sqrt{\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)}\,\left(2n+1+\left|\tilde{m}_{5}\pm\frac{1}{2}\right|\right)$
(49)

with $n=0,1,2,\dots$. If $\tilde{m}_{5}\geq 1/2$, then the spectrum reduces to

$\displaystyle M_{n\frac{L}{R}}^{2>}$
$\displaystyle=(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2})\,k+2\,|k|\,\sqrt{\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)}\,\left(2n+1+\tilde{m}_{5}\pm\frac{1}{2}\right)$
(50a) and if $\tilde{m}_{5}\leq-1/2$, it gives $\displaystyle
M_{n\frac{L}{R}}^{2<}$
$\displaystyle=(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2})\,k+2\,|k|\,\sqrt{\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)}\,\left(2n+1-\tilde{m}_{5}\mp\frac{1}{2}\right).$
(50b)

In the case where $\tilde{m}_{5}$ belongs to the interval $(-1/2,1/2)$, the
spectrum is given by Eq. (III.1). So, we see that, in general, the masses for
the $L$ and $R$ sectors in the deformed AdS model, in this approximation, are
different. In Section V, we are going to see how the masses in the $L$ and $R$
sectors can be related.

##  IV Comment on the Signs of the Dilaton for Fermions

Initially, in this section, before dealing with the question of the signs of
the dilaton for fermions in soft wall models, for clarity, we briefly review
the case for vector mesons.

The soft wall was proposed in Ref. Karch:2006pv  for vector mesons, where the
question of the sign of the dilation was touched and further discussed in Ref.
Karch:2010eg , rewriting the action as

$S=\int dz\,d^{4}{x}\sqrt{-g}\,e^{-az^{2}}\,\mathcal{L},$ (51)

where it as shown that the case $a<0$ implies a spurious massless mode in the
vector meson spectra, in contrast with the more interesting case $a>0$ where
there are only massive physical modes.

On the other side, in order to obtain a confining quark-antiquark potential,
the authors of Ref. Andreev:2006ct  introduced an exponential in the AdS
metric as

$ds^{2}=\frac{\Omega^{2}}{z^{2}}\,h_{2}(z)\,(dz^{2}+dx^{i}dx^{i});\qquad
h_{2}(z)=e^{\frac{1}{2}c_{2}z^{2}},$ (52)

where the constant $c_{2}>0$. This result can generalized to a metric of the
form Bruni:2018dqm

$ds^{2}=\frac{\Omega^{2}}{z^{2}}\,h_{n}(z)\,(dz^{2}+dx^{i}dx^{i});\qquad
h_{n}(z)=e^{\frac{1}{n}(c_{n}z)^{n}},$ (53)

with the restriction $n>0$ besides $c_{n}>0$, so that a linear confining
quark-antiquark potential is also obtained in this case.

It is interesting to observe that an alternative soft wall model for vector
mesons starting with metric (52) and no dilaton in the action is equivalent to
the original soft wall model (51) with $a<0$, which is problematic in the
vector meson spectra.

Now, we comment on the signs of the dilaton for the fermionic models discussed
in Sections II and III above.

First of all, it is important to remark that the sign of the dilaton in the
original and dressed soft wall models for fermions (Section II) is irrelevant,
since it does not modify the spectra or any other of their properties. This is
so, because the exponential of the dilaton field in the fermionic action can
be factored out of the corresponding equations of motion. For this reason, a
potential similar to the dilaton field is added in the dressed model to the
bare fermionic bulk mass, so that these particles acquire a discrete mass
spectrum.

On the other hand, in Section III, the positive sign of quadratic exponential
deformation of the metric, Eqs. (26) and (27), comes from a numerical fit of
this model for baryons of spin 1/2, 3/2 and 5/2 to describe conveniently their
spectra. In fact, this deformed AdS model when applied to glueballs of even
and odd spins also have a positive sign for the quadratic exponential
deformation of the metric, as in the case of fermions. The exceptions are the
cases of scalar and vector mesons for which the deformed AdS model is better
described by a quadratic exponential deformation with negative sign
FolcoCapossoli:2019imm . This is why in Section III, we used the quadratic
exponential deformation with a positive sign.

In view of these observations, we see that the constants $\kappa$, Eq. (3),
and $\sqrt{k}$, Eq. (27), play the role of IR scales in the dressed soft wall
and the deformed AdS models for fermions, in Sections II and III,
respectively. Actually, in Section V below, we present a numerical comparison
of the baryonic spectra obtained from these models where the values of these
constants are of the same order, in the range 200-500 MeV, and in Section VI,
we are going to relate these IR scales of both models when the effective
potential of the deformed AdS model is quadratically approximated.

##  V Numerical comparison of the models

It is interesting to calculate masses within these models for spin 1/2
hadrons, to compare the outputs. Then, taking the case of $N(1/2^{+})$ baryons
the masses were calculated using the dressed soft wall, corresponding to Eqs.
(19), (22), and (24), with $\kappa=$ 271 MeV, $\kappa=$ 500 MeV, and $\kappa=$
470 MeV, respectively, using 939 MeV as inputs. The masses for the deformed
AdS model, were obtained solving numerically Eq.(36) with $k=205^{2}$ MeV2.
These masses are presented in Table 1 together with experimental data coming
from PDG Workman:2022ynf .

| Baryons $N(1/2^{+}$)
---|---
|  $N$ baryon |  $M_{\rm exp}$ | Eq.(19) | Eq.(22) | Eq.(24) | Eq.(36)
$n=0$ |  $N(939)$ | 939 | 939 |  939 | 939 |  987
$n=1$ |  $N(1440)$ | 1370 |  1085 | 1372 | 1330 |  1264
$n=2$ |  $N(1710)$ | 1700 |  1213 | 1698 | 1630 |  1531
$n=3$ |  $N(1880)$ | 1860 |  1329 | 1970 | 1880 |  1791
$n=4$ |  $N(2100)$ | 2100 |  1435 | 2209 | 2100 |  2046
$n=5$ |  $N(2300)$ | 2300 | 1534 | 2425 |  2300 | 2296
Table 1: Masses of $N(1/2^{+})$ baryons expressed in MeV. The column
$n=0,\,1,\,2,\,\dots$ represents the holographic radial quantum number
excitation. The column $M_{\rm exp}$ represents the experimental data coming
from PDG Workman:2022ynf . The columns Eqs. (19), (22), and (24), give the
spectra corresponding to these equations in the dressed soft wall, discussed
in Section II, with $\kappa=$ 271 MeV, $\kappa=$ 500 MeV, and $\kappa=$ 470
MeV, respectively, using 939 MeV as inputs. The column Eq.(36) represents the
masses obtained numerically within the deformed AdS model, Section III, with
$k=205^{2}$ MeV2.

One can note from Table 1 that the masses obtained for the dressed soft wall
model, starting from 939 MeV, from Eqs. (19), (22), and (24), vary
significantly, depending on additional modeling as the inclusion of hyperfine
spin-spin and meson cloud interactions, Eq. (22), or anomalous dimensions, Eq.
(24). In particular, one sees from the column of Table 1, corresponding to Eq.
(19), that these masses are between 20 and 33$\%$ lower than the experimental
data from PDG. After including hyperfine spin-spin and meson cloud
interactions, as in Ref. Gutsche:2011vb , one obtains from Eq. (22), very good
results, specially for $n=1,2$, with difference with PDG lower than 0.15$\%$,
while for $n=3,4,5$ the masses are higher than that of PDG by 5 to 6$\%$.
Still in the dressed soft wall model, this time with anomalous dimensions, the
masses given by Eq. (24). For $n=1,2$ they are lower than that of PDG by 3 to
4$\%$, while for $n=3,4,5$ they are very good compared with experimental data.
Finally, from the deformed AdS model, the masses coming from the numerical
solution to Eq. (36), are all lower than that of PDG, for $n=1,2$, with
deviation of 8 and 10$\%$, respectively, while for $n=3,4,5$, with 3.7, 2.6
and 0.2$\%$, respectively. Note that in this case, the mass for the nucleon
ground state is 5.1$\%$ higher than the well known value of 939 MeV.

##  VI Analytic comparison of the models

The models we described above have different approaches to spin 1/2 fields in
the $AdS_{5}$ or asymptotically $AdS_{5}$ spaces. The deformed background
space model presents a warp factor in the metric, with no extra factor in the
action. Conversely, the dressed soft wall model presents no deformations in
the metric, but the addition of a $z^{2}$ dependent factor in the mass term of
the action, besides the dilaton field.

In particular, when applied to fermions we see that they lead to distinct
equations of motion. In the dressed soft wall model we have

$\left[-\partial_{z}^{2}+\kappa^{4}\,z^{2}+2\,\kappa^{2}\left(m_{5}\mp\frac{1}{2}\right)+\frac{m_{5}(m_{5}\pm
1)}{z^{2}}\right]{f}^{n}_{\frac{L}{R}}(z)=M_{n}^{2}\,{f}^{n}_{\frac{L}{R}}(z),$
(54)

while in the deformed AdS model we get

$\left\\{-\partial^{2}_{\tilde{z}}+\tilde{m}_{5}\,e^{\,A(\tilde{z})}\left[\tilde{m}_{5}\,e^{\,A(\tilde{z})}\mp\,A^{\prime}(\tilde{z})\right]\right\\}\,\psi_{\frac{L}{R}}^{n}(\tilde{z})=M_{n}^{2}\,\psi^{n}_{\frac{L}{R}}(\tilde{z}),$
(55)

which, in the quadratic approximation, reads

$\left[-\partial^{2}_{\tilde{z}}+\tilde{m}_{5}\left(\frac{\tilde{m}_{5}}{2}\mp\frac{3}{8}\right)\,k^{2}\,\tilde{z}^{2}+\tilde{m}_{5}\left(\tilde{m}_{5}\mp\frac{1}{2}\right)\,k+\frac{\tilde{m}_{5}\,(\tilde{m}_{5}\pm
1)}{\tilde{z}^{2}}\right]\,\psi_{\frac{L}{R}}^{n}(\tilde{z})=M_{n}^{2}\,\psi_{\frac{L}{R}}^{n}(\tilde{z}).$
(56)

Comparing Eqs. (54) and (56), we see that they have the same structure, that
is, a harmonic and a centrifugal/centripetal barrier term plus a constant.
Both have similar solutions in terms of the associated Laguerre polynomials
(see Appendix A). The equivalence of the left and right modes of Eqs. (54) and
(56) hold, if and and only if the following conditions are satisfied:

$\displaystyle m_{5}\,(m_{5}\pm 1)$
$\displaystyle=\tilde{m}_{5}\,(\tilde{m}_{5}\pm 1),$ (57a) $\displaystyle
2\,\kappa^{2}\,\left(m_{5}\mp\frac{1}{2}\right)$
$\displaystyle=\tilde{m}_{5}\,\left(\tilde{m}_{5}\mp\frac{1}{2}\right)\,k,$
(57b) $\displaystyle\kappa^{4}$
$\displaystyle=\tilde{m}_{5}\,\left(\frac{\tilde{m}_{5}}{2}\mp\frac{3}{8}\right)\,k^{2}.$
(57c)

Let us now look at what these conditions imply.

First of all, equation (57a) imply one of the following conditions:

$\displaystyle(i)$ $\displaystyle\qquad\qquad m_{5}=\tilde{m}_{5},$ (58a)
$\displaystyle(ii)$ $\displaystyle\qquad\qquad m_{5}=-\tilde{m}_{5}\mp 1.$
(58b)

We now study the the consistency of these solutions with the other two
conditions, Eqs. (57b) and (57c).

Let us discuss here the moist interesting case $(i)$ where
$m_{5}=\tilde{m}_{5}$, and leave the other to Appendix B. Imposing the
condition $m_{5}=\tilde{m}_{5}$ into Eq. (57b), one gets

$\displaystyle 2\,\kappa^{2}=\tilde{m}_{5}\,k.$ (59)

Then, substituting this equation into Eq. (57c), one finds the relations

$\displaystyle m_{5\pm}=\tilde{m}_{5\pm}=\pm\frac{3}{2},\qquad\qquad{\rm
and}\qquad\qquad\,k_{\pm}\,=\pm\frac{4}{3}\kappa^{2},$ (60)

where the plus (minus) signs refer to the left (right) modes and imply
relations between the mass parameters and scales of the dressed soft wall and
deformed AdS models.

Since $\kappa$ is real, one concludes that the conditions, Eqs. (57a), (57b),
and (57c), imply that the IR scales $k_{\pm}$ of the quadratic approximation
for the deformed AdS model must have different signs for its left $(+)$ and
right $(-)$ modes. Also, the left modes should have mass parameters
$m_{5+}=\tilde{m}_{5+}=+3/2$, while for the right modes
$m_{5-}=\tilde{m}_{5-}=-3/2$, in both models.

These differences between the left and right modes of both models suggest that
they are not equivalent even in the quadratic approximation considered for the
deformed AdS case. This non-equivalence of the models will be confirmed by the
discussion of the spectra below.

###  VI.1 Spectra

Now, to compare the spectra of these analytical models, consider the effective
Schrödinger equation for the dressed soft wall, Eq. (54). Using
$m_{5\pm}=\tilde{m}_{5\pm}=\pm 3/2$ and
$\,k_{\pm}\,=\pm\frac{4}{3}\kappa^{2}$, one finds the equations

$\left(-\partial^{2}_{z}+\frac{9}{16}\,(k_{\pm})^{2}\,z^{2}+\frac{3}{2}\,k_{\pm}+\frac{15}{4}\,\frac{1}{z^{2}}\right)f^{n}_{L/R}(z)=M^{2}_{n}\,f^{n}_{L/R}(z),$
(61)

for the left $(+)$ and right $(-)$ modes. These equations are also obtained
when the relations (60) are substituted into the effective Schrödinger
equations for the deformed AdS model, Eq. (56), with $z$ replaced by
$\tilde{z}$, for both the left and right modes.

The term proportional to $(k_{\pm})^{2}$ is the same for the left and right
modes, together with the centrifugal barrier, assuring a potential well
bounded from below in both cases. However, the term proportional to $k_{\pm}$
have opposite signs implying a shift in the energy between the left and right
modes.

The spectrum for the left modes, obtained from Eq. (61) with
$m_{5+}=\tilde{m}_{5+}=3/2$, and $k_{+}=(4/3)\,\kappa^{2}$, in both models
(from Eqs. (15), and (50a), upper signs), is the following:

$M_{nL}^{2}=3k_{+}\,(n+2)=4\kappa^{2}(n+2);\qquad\qquad(n=0,1,2,\dots).$ (62)

On the other hand, the spectrum for the right modes, also obtained from Eq.
(61), now with $m_{5-}=\tilde{m}_{5-}=-3/2$ and $k_{-}=-(4/3)\,\kappa^{2}$, in
both models (from Eqs. (16), and (50b), lower signs), is:

$M_{nR}^{2}=3|k_{-}|\,(n+1)=4\kappa^{2}(n+1);\qquad\qquad(n=0,1,2,\dots).$
(63)

Note that the spectrum for the left and right modes, Eqs. (62) and (63),
respectively, are different due to the energy shift caused by the term
$(3/2)k_{\pm}$. Actually, they are almost degenerate, except for the ground
states.

So, clearly, the above spectra analysis shows that the dressed soft wall and
the deformed AdS models for fermions, despite some similarities, are not
equivalent, even in the quadratic approximation of the latter.

###  VI.2 Wave Functions

In order to obtain a more complete picture in the comparison of the dressed
soft wall and the quadratic approximation for the deformed AdS models, here we
obtain the analytic wave functions for both.

As shown in the previous section, for $m_{5\pm}=\tilde{m}_{5\pm}=\pm 3/2$, one
finds the one dimensional equations of motion in both models in terms of the
holographic coordinate $\zeta$

$\left(-\partial^{2}_{\zeta}+\frac{9}{16}\,(k_{\pm})^{2}\,\zeta^{2}+\frac{3}{2}\,k_{\pm}+\frac{15}{4}\,\frac{1}{\zeta^{2}}\right)f^{n}_{L/R}(\zeta)=M^{2}_{n}\,f^{n}_{L/R}(\zeta),$
(64)

for the left and right modes, respectively, or equivalently

$\left(-\partial^{2}_{\zeta}+\,\kappa^{4}\,\zeta^{2}\pm
2\,\kappa^{2}+\frac{15}{4}\,\frac{1}{\zeta^{2}}\right)f^{n}_{L/R}(\zeta)=M^{2}_{n}\,f^{n}_{L/R}(\zeta),$
(65)

where $\zeta$ stands for $z$ in the dressed soft wall and for $\tilde{z}$ in
the deformed AdS models. Then, one finds that the normalizable solutions for
the left and right modes are given exactly by the same expression in terms of
$\zeta$ in both models

$\displaystyle{f}^{n}_{{}_{L/R}}(\zeta)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma(n+3)}}\,|\kappa|^{3}\,\zeta^{5/2}\,e^{-\kappa^{2}\zeta^{2}/2}\,L^{2}_{n}(\kappa^{2}\zeta^{2}),$
(66a)

for $n=0,1,2,\dots$, since the terms $\pm 2\,\kappa^{2}$ only produce a shift
in the energy of the modes.

So, the wave functions obtained for the dressed soft wall and the quadratic
approximation for the deformed AdS have the same dependence on $\zeta$.
However, since $\zeta$ is identified with $z$ in the dressed soft wall and as
$\tilde{z}$ in the deformed AdS model, actually these solutions are not
equivalent. In the UV region in both models ($\zeta\to 0$) this difference
vanishes, but in the IR ($\zeta\to\infty$) this difference is relevant.

###  VI.3 Comments on the DIS in both models

In this section, we briefly comment on the DIS obtainable from the dressed
soft wall and the quadratic approximation of the deformed AdS model. The
holographic approach to DIS starts with an interaction action which is an
integral over the virtual photonic and the baryonic initial and final states.
From this action it is possible to obtain, for instance, the structure
functions $F_{1}(x,q)$ and $F_{2}(x,q)$, where $x$ is the momentum fraction of
the parton, also called the Bjorken variable, defined as $x=q^{2}/2p\cdot q$,
with $q^{\mu}$ and $p^{\mu}$ the momenta of the virtual photon and of the
initial baryon, respectively Polchinski:2002jw ; BallonBayona:2007qr ;
Braga:2011wa ; FolcoCapossoli:2020pks .

The relevant interaction action for DIS in the dressed soft wall model is
given by BallonBayona:2007qr ; Braga:2011wa

$S_{int}=g_{V}\int
dz\,d^{4}x\,\sqrt{-g}\,e^{-\kappa^{2}z^{2}}\,\frac{z}{\Omega}\,A_{m}\bar{\Psi}_{X}\,\gamma^{m}\,\Psi_{i},$
(67)

where $g_{V}$ is the coupling constant of the interaction of the
electromagnetic field $A_{m}=A_{m}(x,z)$ with the initial
$\Psi_{i}=\Psi_{i}(x,z)$ and final $\Psi_{X}=\Psi_{X}(x,z)$ fermion states.
The fermionic states are given by Eq. (7), while the electromagnetic field is
written in terms of the confluent hypergeometric function of the second kind
BallonBayona:2007qr ; Braga:2011wa :

$\displaystyle A_{\mu}(x^{\mu},z)$ $\displaystyle=$
$\displaystyle\eta_{\mu}e^{iq\cdot
x}\kappa^{2}z^{2}\,\Gamma\left(1+\frac{q^{2}}{4\kappa^{2}}\right)\,{\cal
U}\left(1+\frac{q^{2}}{4\kappa^{2}};2;\kappa^{2}z^{2}\right),$ (68a)
$\displaystyle A_{z}(x^{\mu},z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\eta\cdot q\,z\,e^{iq\cdot
x}\,\Gamma\left(1+\frac{q^{2}}{4\kappa^{2}}\right)\,\,{\cal
U}\left(1+\frac{q^{2}}{4\kappa^{2}};1;\kappa^{2}z^{2}\right),$ (68b)

where $q_{\mu}$ and $\eta_{\mu}$ are the momentum and polarization of the
virtual photon, respectively.

On the other side, the DIS interaction action in the deformed AdS model is
given by FolcoCapossoli:2020pks

${\tilde{S}}_{int}=\tilde{g}_{V}\int
d\tilde{z}\,d^{4}x\,\sqrt{-\tilde{g}}\,\frac{\tilde{z}}{\tilde{\Omega}}\,\tilde{A}_{m}\tilde{\bar{\Psi}}_{X}\,\gamma^{m}\,\tilde{\Psi}_{i},$
(69)

where the fermionic fields are given by Eq. (34), whilst the gauge fields are
described by

$\displaystyle\tilde{A}_{\mu}(x^{\mu},\tilde{z})$ $\displaystyle=$
$\displaystyle-\eta_{\mu}e^{iq\cdot
x}\frac{k}{2}\tilde{z}^{2}\,\Gamma\left(1-\frac{q^{2}}{2k}\right)\,{\cal
U}\left(1-\frac{q^{2}}{2k};2;-\frac{k}{2}\tilde{z}^{2}\right),$ (70a)
$\displaystyle\tilde{A}_{\tilde{z}}(x^{\mu},\tilde{z})$ $\displaystyle=$
$\displaystyle\frac{i}{2}\eta\cdot q\,\tilde{z}\,e^{iq\cdot
x}\,\Gamma\left(1-\frac{q^{2}}{2k}\right)\,\,{\cal
U}\left(1-\frac{q^{2}}{2k};1;-\frac{k}{2}\tilde{z}^{2}\right).$ (70b)

Since $k$ is considered negative for the photon fields in this model (see
MartinContreras:2021yfz  for details), these solutions have the same
dependence on the holographic coordinate $\zeta$ (introduced in Eqs. (64) and
(65)) as those presented for the dressed soft wall model, Eqs. (68a) and
(68b). This is also the case of the fermionic fields discussed in the previous
section.

Further, we see that there are additional differences in the description of
the DIS in the interaction actions, Eqs. (67) and (69). Clearly, the presence
of the dilaton in action of the dressed soft wall and the warp factor in the
deformed AdS model bring in some distictions between them, although the
structure functions that can be obtained from these interaction actions in the
dressed soft wall and the (full) deformed AdS models are comparable, as can be
seen in Refs. Braga:2011wa  and FolcoCapossoli:2020pks .

##  VII Conclusions

In this work, we have studied the dressed soft wall and the deformed AdS
models for fermions and compared their effective Schrödinger equations and
spectra. This comparison was motivated by recent results on proton DIS
structure functions obtained from the deformed AdS model
FolcoCapossoli:2020pks  which resemble that of the dressed soft wall model
Braga:2011wa . These two models have in common a quadratic exponential factor,
which in the case of the dressed soft wall is included in the action and in
the mass term of the AdS bulk fermions. On the other hand, the deformed AdS
model contains this exponential factor just in the metric which is
asymptotically AdS in the UV. In particular for fermions, the dressed soft
wall model is analytically solvable, while the deformed AdS model needs
numerical solutions.

In Section II, we studied the dressed soft wall model for fermions and
obtained its effective Schrödinger equations for the left and right modes and
their solutions in terms of associated Laguerre functions. Then, we obtained
the corresponding fermionic spectra including some modifications as the
hyperfine spin-spin and meson cloud interactions as in Ref. Gutsche:2011vb ,
as well as anomalous dimensions as considered in Ref. FolcoCapossoli:2020pks .

In Section III, we discussed the deformed AdS model for fermions and obtained
its effective Schrödinger equations for the left and right modes for which
numerical solutions were found in FolcoCapossoli:2019imm ;
FolcoCapossoli:2020pks . So, in order to make a comparison with the dressed
soft wall model, we perform a power expansion of the effective exponential of
the deformed model and consider a quadratic approximation which leads to
analytical solutions given by the associated Laguerre functions with
corresponding spectra. In Section IV, we comment on the differences between
the choices of the sign of the dilaton in both models. In Section V, we
present a numerical comparison of the various spectra associated with the
dressed soft wall model, discussed in Section II, and the numerical spectra
for the deformed AdS model for $N(1/2^{+})$ baryons.

In Section VI, we present a comparison between the dressed soft wall model and
the quadratic approximation for the deformed AdS model, discussed in Sections
II and III. These two models are analytically solvable for fermions with
solutions given by the associated Laguerre functions, although they are not
exactly equivalent. Actually, we find relations between these two models and
obtain the necessary conditions to relate the left and right modes for each
case. We find that when $m_{5}=\tilde{m}_{5}$, we discover that
$m_{5}=\tilde{m}_{5}=\pm 3/2$, where the upper (lower) sign refers to the left
(right) modes. This means that to fully describe the left and right modes in
the dressed soft wall model from the quadratic approximation of the deformed
model one needs to consider fermionic bulk masses with opposite signs. It is
interesting to note that the AdS/CFT correspondence implies a relation between
the fermionic bulk dimensionless parameter $m_{5}$ and the conformal dimension
$\Delta$ of the dual boundary operator: $|m_{5}|=\Delta-2$, so that the left
and right modes discussed here with different signs for $m_{5}$ and
$\tilde{m}_{5}$ actually are dual to operators with the same conformal
dimension $\Delta$. It is also important to mention that the parameters
$m_{5}$ and $\tilde{m}_{5}$ are not the fermionic bulk massses, but are
related to these masses as $m_{5}=\mu_{5}\Omega$ and
$\tilde{m}_{5}=\tilde{\mu}_{5}\tilde{\Omega}$, where $\Omega$ and
$\tilde{\Omega}$ are the AdS radii in the two models, which remain arbitrary.
So, in some sense, one has some freedom in establishing the relations between
the two models. We have also shown that the analytical wave functions of both
models have the same mathematical description in terms of the holographic
coordinate $zeta$, which is identified with $z$ for the dressed soft wall and
$\tilde{z}$ for the deformed AdS model. Finally, we comment on the DIS
description of both models and show that they have many similarities but are
not mathematically equivalent.

The study presented here could help understand better structure functions and
form factors, as discussed, for instance, in Refs. Abidin:2009hr ;
Mamo:2021cle ; Mamo:2021jhj ; Braga:2011wa ; FolcoCapossoli:2020pks ;
Contreras:2021epz . This is currently under investigation.

## Acknowledgments

We would like to thank Kostas Rigatos for discussions and an anonymous referee
for suggestions. A.C.P.N. is supported by Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES). H.B.-F. is partially supported by Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under grant $\\#$
311079/2019-9.

##  Appendix A The isotonic oscillator

In this Appendix, we collect the basic results needed in this work from the
quantum mechanical isotonic oscillator, described by the Schrödinger equation
(see, e. g., Ikhdair )

$\left(-\partial^{2}_{z}-\epsilon_{n}+\,\beta^{2}\,z^{2}+\frac{\alpha}{z^{2}}\right)\psi_{n}(z)=0,$
(71)

which for $z>0$, $\beta>0$, and $\alpha>-\frac{1}{4}$ gives the spectrum

$\epsilon_{n}=2\,\beta\,\left(2n+1+\xi\right);\qquad(n=0,1,2,\dots),$ (72)

where $\xi=\frac{1}{2}\sqrt{1+4\alpha}>0$, and imply the normalized wave
functions

$\displaystyle\psi_{n}(z)=\sqrt{\frac{2\,\beta^{\xi+1}n!}{\Gamma(n+\xi+1)}}\,z^{\xi+1/2}\,e^{-\beta
z^{2}/2}\,L^{\xi}_{n}(\beta z^{2}).$ (73)

##  Appendix B Case $(ii)$: $m_{5}=-\tilde{m}_{5}\mp 1$

In this Appendix, we analyse the case $m_{5}=-\tilde{m}_{5}\mp 1$, Eq. (57a).
Inserting this relation in Eq. (57b), one gets

$\displaystyle
2\,\kappa^{2}\,\left(-\tilde{m}_{5}\mp\frac{3}{2}\right)=\tilde{m}_{5}\,\left(\tilde{m}_{5}\mp\frac{1}{2}\right)\,k.$
(74)

Then, squaring this equation and substituting in it the ratio
$\kappa^{4}/k^{2}$ from Eq. (57c), one obtains the cubic equation

$(\tilde{m}_{5})^{3}\pm\frac{11}{2}(\tilde{m}_{5})^{2}-\frac{1}{4}\tilde{m}_{5}\mp\frac{27}{8}=0,$
(75)

which has real roots

$\tilde{m}_{5\frac{L}{R}}^{(1)}=\mp
5.43163;\qquad\tilde{m}_{5\frac{L}{R}}^{(2)}=\mp
0.82319;\qquad\tilde{m}_{5\frac{L}{R}}^{(3)}=\pm 0.75482,$ (76)

with the corresponding ratios

$\left(\frac{\kappa^{2}}{k}\right)_{\frac{L}{R}}^{(1)}=\pm
4.09734;\qquad\left(\frac{\kappa^{2}}{k}\right)_{\frac{L}{R}}^{(2)}=\mp
0.804684;\qquad\left(\frac{\kappa^{2}}{k}\right)_{\frac{L}{R}}^{(3)}=\mp
0.0426516.$ (77)

This way, from the effective Schrödinger equations for the dressed soft wall,
Eq. (54), we obtain that these solutions satisfy

$\left[-\partial_{z}^{2}+\kappa^{4}\,z^{2}-2\,\kappa^{2}\left(\tilde{m}_{5}\pm\frac{3}{2}\right)+\frac{\tilde{m}_{5}(\tilde{m}_{5}\pm
1)}{z^{2}}\right]{f}^{n}_{L/R}(z)=M_{n}^{2}\,{f}^{n}_{L/R}(z),$ (78)

with mass spectrum

$M_{nL/R}^{2}=\kappa^{2}\left[4n+2+\left|2\tilde{m}_{5}+1\right|-2\tilde{m}_{5}\mp
3\right];\qquad(n=0,1,2,\dots).$ (79)

Note that all values of $\tilde{m}^{(i)}_{L/R}$ given by Eq. (76), satisfy the
stability condition of the normalizable solutions,
$\tilde{m}_{5}(\tilde{m}_{5}\pm 1)>-1/4$ (see Appendix A).

On the other side, the effective Schrödinger equations coming from the
deformed AdS model, Eq. (56), read

$\left[-\partial^{2}_{z}+\tilde{m}_{5}\left(\frac{\tilde{m}_{5}}{2}\mp\frac{3}{8}\right)\,k^{2}\,z^{2}+\tilde{m}_{5}\left(\tilde{m}_{5}\mp\frac{1}{2}\right)\,k+\frac{\tilde{m}_{5}\,(\tilde{m}_{5}\pm
1)}{z^{2}}\right]\,\psi_{L/R}^{n}(z)=M_{n}^{2}\,\psi_{L/R}^{n}(z).$ (80)

which are equivalent to Eq. (78) for the masses given by (76) and ratios Eq.
(77). The corresponding spectra are given by Eq. (50a) for $\tilde{m}_{5}>1/2$

$\displaystyle M_{nL/R}^{2>}$
$\displaystyle=(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2})\,k+2\,|k|\,\sqrt{\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)}\,\left(2n+1+\tilde{m}_{5}\pm\frac{1}{2}\right),$
(81a) and Eq. (50b), for $\tilde{m}_{5}\leq-1/2$ $\displaystyle M_{nL/R}^{2<}$
$\displaystyle=(\tilde{m}_{5}^{2}\mp\frac{\tilde{m}_{5}}{2})\,k+2\,|k|\,\sqrt{\left(\frac{\tilde{m}_{5}^{2}}{2}\mp\frac{3\,\tilde{m}_{5}}{8}\right)}\,\left(2n+1-\tilde{m}_{5}\mp\frac{1}{2}\right),$
(81b)

for $\tilde{m}_{5}$ and the ratios $\kappa^{2}/k$ obeying Eqs. (76) and (77),
respectively.

So, clearly, also in the case $m_{5}=-\tilde{m}_{5}\mp 1$, this spectra
analysis shows that the dressed soft wall and the deformed AdS models for
fermions, are not equivalent, even in the quadratic approximation of the
latter.

##  Appendix C Comment on relations with top-down models

Fermionic top-down models starting with type II-A or II-B string theory with
various configurations with D$p$-branes lead to analytical solutions of the
form Kirsch:2006he ; Abt:2019tas ; Nakas:2020hyo

$\displaystyle\psi_{\cal G}$ $\displaystyle=$
$\displaystyle\frac{\varrho^{\ell+1}}{(1+\varrho^{2})^{(n+\ell+\frac{m}{2}+\frac{5}{4})}}\,_{2}F_{1}\left(-n,-n-\ell-\frac{m+1}{2};\ell+\frac{m+3}{2};-\varrho^{2}\right)\chi_{+}$
(83)
$\displaystyle+\frac{\varrho^{\ell}}{(1+\varrho^{2})^{(n+\ell+\frac{m}{2}+\frac{5}{4})}}\,_{2}F_{1}\left(-n,-n-\ell-\frac{m+3}{2};\ell+\frac{m+1}{2};-\varrho^{2}\right)\chi_{-},$
$\displaystyle\psi_{\cal F}$ $\displaystyle=$
$\displaystyle\frac{\varrho^{\ell}}{(1+\varrho^{2})^{(n+\ell+\frac{m}{2}+\frac{1}{4})}}\,_{2}F_{1}\left(-n,-n-\ell-\frac{m-1}{2};\ell+\frac{m+1}{2};-\varrho^{2}\right)\chi_{+}$
(85)
$\displaystyle+\frac{\varrho^{\ell+1}}{(1+\varrho^{2})^{(n+\ell+\frac{m}{2}+\frac{1}{4})}}\,_{2}F_{1}\left(-n,-n-\ell-\frac{m+1}{2};\ell+\frac{m+3}{2};-\varrho^{2}\right)\chi_{-},$

which depend on the Gauss hypergeometric function

${}_{2}F_{1}(a,b;c;z)=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{k!(c)_{k}}z^{k},$
(86)

with spectra

$\displaystyle\bar{M}_{\cal
G}^{2}=4\left(n+\ell+\frac{m+1}{2}\right)\left(n+\ell+\frac{m+3}{2}\right),$
(87) $\displaystyle\bar{M}_{\cal
F}^{2}=4\left(n+\ell+\frac{m-1}{2}\right)\left(n+\ell+\frac{m+1}{2}\right).$
(88)

On the other side, the associate Laguerre functions in terms of which we
described the analytical solutions of the fermionic soft walls models
discussed in this work can be written as Abramowitz

$L_{n}^{(\alpha)}(x)=\frac{\Gamma(\alpha+n+1)}{n!\Gamma(\alpha+1)}\,_{1}F_{1}(-n;\alpha+1;x),$
(89)

in terms of the Kummer confluent hypergeometric function

${}_{1}F_{1}(a;c;z)=\sum_{k=0}^{\infty}\frac{(a)_{k}}{k!(c)_{k}}z^{k},$ (90)

where

$\displaystyle(x)_{k}$ $\displaystyle\equiv$
$\displaystyle\Gamma(x+k)/\Gamma(x)$ (91) $\displaystyle=$ $\displaystyle
x(x+1)\cdots(x+k-1)$ (92)

are the Pochhmammer symbols.

Then, in the particular case where $(b)_{k}=1$, the two problems are directly
related. This happens when $b=0$, which imply

$\displaystyle 0=n+\ell+\frac{m+1}{2}=n+\ell+\frac{m+3}{2};\qquad({\cal G}),$
(93) $\displaystyle 0=n+\ell+\frac{m-1}{2}=n+\ell+\frac{m+1}{2};\qquad({\cal
F}),$ (94)

meaning massless D$p$-brane modes

$\displaystyle\bar{M}^{2}_{\cal G}=\bar{M}^{2}_{\cal F}=0.$ (95)

So, we see that the dressed soft wall and the quadratic approximation for the
deformed AdS models discussed here seem to have fewer degrees of freedom than
the D$p$-brane models of Refs. Kirsch:2006he ; Abt:2019tas ; Nakas:2020hyo .

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