# Phase transitions and light scalars in bottom-up holography

Daniel Elander  Laboratoire Charles Coulomb (L2C), University of Montpellier,
CNRS, Montpellier, France  Ali Fatemiabhari
<EMAIL_ADDRESS>Department of Physics, Faculty of Science
and Engineering, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales,
UK  Maurizio Piai  Department of Physics, Faculty of Science and Engineering,
Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, UK

###### Abstract

Within the bottom-up approach to holography, we construct a class of six-
dimensional gravity models, and discuss solutions that can be interpreted,
asymptotically in the far UV, in terms of dual five-dimensional conformal
field theories deformed by a single scalar operator. We treat the scaling
dimension of such operator, related to the mass of the one scalar field in the
gravity theory, as a free parameter. One dimension in the regular geometry is
compactified on a shrinking circle, hence mimicking confinement in the
resulting dual four-dimensional theories.

We study the mass spectrum of bosonic states. The lightest state in this
spectrum is a scalar particle. Along the regular (confining) branch of
solutions, we find the presence of a tachyonic instability in part of the
parameter space, reached by a smooth deformation of the mass spectrum, as a
function of the boundary value of the background scalar field in the gravity
theory. In a region of parameter space nearby the tachyonic one, the lightest
scalar particle can be interpreted as an approximate dilaton, sourced by the
trace of the stress-energy tensor, and its mass is parametrically suppressed.

We also compute the free energy, along several branches of gravity solutions.
We find that both the dilatonic and tachyonic regions of parameter space,
identified along the branch of confining solutions, are hidden behind a first-
order phase transition, so that they are not realised as stable solutions,
irrespectively of the scaling dimension of the deforming field-theory
operator. The (approximate) dilaton, in particular, appears in metastable
solutions. Yet, the mass of the lightest state, computed close to the phase
transition, is (mildly) suppressed. This feature is amplified when the (free)
parameter controlling the scaling dimension of the deformation is $5/2$, half
the dimension of space-time in the field theory.

###### Contents

  1. I Introduction
  2. II The model
    1. II.1 Action in six dimensions
    2. II.2 Dimensional reduction to five dimensions
  3. III Classes of solutions
    1. III.1 UV expansions
    2. III.2 Supersymmetric solutions
    3. III.3 Confining solutions
    4. III.4 Singular domain-wall solutions
  4. IV Mass spectrum of fluctuations
  5. V Free energy
  6. VI Summary
  7. VII Outlook
  8. A Sigma-model coupled to gravity
    1. A.1 Linearised equations for scalar fluctuations
  9. B Of superpotentials
  10. C Asymptotic expansion of the background solutions
  11. D Asymptotic expansions of the fluctuations
    1. D.1 IR expansions
    2. D.2 UV expansions
  12. E Probe approximation
    1. E.1 IR expansions
    2. E.2 UV expansions
  13. F More about the free energy

##  I Introduction

Investigations into the fundamental nature and origin of the Higgs boson, the
latest particle of the Standard Model (SM) to have been discovered Aad:2012tfa
; Chatrchyan:2012xdj , are among the most topical in theoretical physics, in
view of the ongoing precision experimental programme at the Large Hadron
Collider. Because of the approximate (explicitly and spontaneously broken)
scale invariant nature of the SM theory, the Higgs boson itself is in fact a
dilaton, the pseudo-Nambu-Goldstone boson associated with dilatations. It is
natural to question whether this is just an accidental symmetry in the SM, or
whether it has a more fundamental, important origin and role Goldberger:2008zz
. In particular, is it possible for the Higgs boson to originate as a
composite dilaton in a more fundamental theory? And if so, what type of
underlying dynamics would yield realistic values for its mass and couplings?

The literature on the dilaton effective field theory dates back many decades
Migdal:1982jp ; Coleman:1985rnk , and its applications have been long
discussed in the context of dynamical electroweak symmetry breaking
Leung:1985sn ; Bardeen:1985sm ; Yamawaki:1985zg , extensions of the SM
Hong:2004td ; Dietrich:2005jn ; Hashimoto:2010nw ; Appelquist:2010gy ;
Vecchi:2010gj ; Chacko:2012sy ; Bellazzini:2012vz ; Bellazzini:2013fga ;
Abe:2012eu ; Eichten:2012qb ; Hernandez-Leon:2017kea , and, more recently, the
interpretation of lattice data Matsuzaki:2013eva ; Golterman:2016lsd ;
Kasai:2016ifi ; Hansen:2016fri ; Golterman:2016cdd ; Appelquist:2017wcg ;
Appelquist:2017vyy ; Golterman:2018mfm ; Cata:2019edh ; Appelquist:2019lgk ;
Cata:2018wzl ; Golterman:2020tdq ; Golterman:2020utm ;
LatticeStrongDynamicsLSD:2021gmp ; Appelquist:2020bqj ; Appelquist:2022qgl ;
Appelquist:2022mjb , in particular in view of the numerical work on certain
$SU(3)$ gauge theories that show indications of a light scalar bound state in
the spectrum Aoki:2014oha ; Appelquist:2016viq ; Aoki:2016wnc ;
Gasbarro:2017fmi ; Appelquist:2018yqe ; Hasenfratz:2022qan ; Fodor:2012ty ;
Fodor:2015vwa ; Fodor:2016pls ; Fodor:2017nlp ; Fodor:2019vmw ; Fodor:2020niv
.

The context of gauge-gravity dualities Maldacena:1997re ; Gubser:1998bc ;
Witten:1998qj ; Aharony:1999ti  is particularly suitable for describing the
dilaton and its dynamical origin, both in its bottom-up Goldberger:1999uk ;
DeWolfe:1999cp ; Goldberger:1999un ; Csaki:2000zn ; ArkaniHamed:2000ds ;
Rattazzi:2000hs ; Kofman:2004tk ; Elander:2011aa ; Kutasov:2012uq ;
Lawrance:2012cg ; Elander:2012fk ; Goykhman:2012az ; Evans:2013vca ;
Megias:2014iwa ; Elander:2015asa  as well as top-down Elander:2009pk ;
Elander:2012yh ; Elander:2014ola ; Elander:2017cle ; Elander:2017hyr
realisations. In what follows, we restrict our attention to holographic duals
in which the geometry of a completely smooth classical background in a higher-
dimensional gravity theory contains a shrinking circle, the shrinking of which
can be used to mimic confinement in the dual field theory, as suggested in
Ref. Witten:1998zw —see also Refs. Sakai:2004cn ; Sakai:2005yt ; Brower:2000rp
; Wen:2004qh ; Kuperstein:2004yf ; Elander:2013jqa ; Elander:2018aub . The
spectra of bound states of the strongly-coupled theory can be computed
perturbatively in its gravity dual, by exploiting the holographic gauge-
invariant formalism developed in Refs. Bianchi:2003ug ; Berg:2005pd ;
Berg:2006xy ; Elander:2009bm ; Elander:2010wd —see also Refs. Elander:2014ola
; Elander:2017cle ; Elander:2017hyr ; Elander:2018aub ; Elander:2020csd . A
useful diagnostic tool to identify (approximate) dilatons within such spectra
is provided by the probe approximation, as discussed in Ref. Elander:2020csd .
Furthermore, one can apply holographic renormalisation Bianchi:2001kw ;
Skenderis:2002wp ; Papadimitriou:2004ap  to compute field-theory quantities,
such as the free energy. We also find it appealing to adopt a simple scale-
setting procedure, such as that proposed in Ref. Csaki:2000cx .

Among all possible realisations of the dilaton scenario, we concentrate on the
ideas developed in Refs. Kaplan:2009kr ; Gorbenko:2018ncu ; Gorbenko:2018dtm ;
Pomarol:2019aae —see also Ref. CruzRojas:2023jhw  and references therein.
Namely, we want to understand whether a light dilaton state might arise in
strongly-coupled theories, the renormalisation group (RG) flow of which brings
them in close proximity of a tachyonic instability. In the holographic dual
description of conformal field theories (CFTs)—fixed points of the RG flow—the
notion of the Breitenlohner-Freedman (BF) bound Breitenlohner:1982jf  captures
the type of instability of interest. In the case of confining theories, one
can generalise this notion, and ultimately arrive to the following, similar
conclusions for physical observables Elander:2020ial ; Elander:2020fmv ;
Elander:2021wkc . In summary, the classical instability is resolved by the
presence of a first-order phase transition. The unstable region of parameter
space cannot be physically realised, which protects the physical spectrum from
developing a tachyon. Nevertheless, one may ask whether, for choices of
parameters close to the phase transition, the lightest (neutral) scalar in the
spectrum has mass and properties that are influenced by the presence of the
tachyonic instability itself, even if the latter is only present in an
unphysical region of parameter space.

The analysis of Refs. Elander:2020ial ; Elander:2020fmv ; Elander:2021wkc
indicates that it is possible to realise this scenario within well-established
examples of top-down holographic models, derived as classical solutions of
supergravities in various dimensions. In all three cases analysed so far, it
has been found that for choices of the parameters that are close to the phase
transition boundary, the lightest scalar state is not parameterically light,
as was predicted in Ref. Pomarol:2019aae , on the basis of the results from a
bottom-up model obtained in a dedicated construction—for models in which no
light dilaton is present, see for instance Refs. Arean:2012mq ; Arean:2013tja
.

In this paper, inspired by Refs. Elander:2020ial ; Elander:2020fmv ;
Elander:2021wkc , we build a class of bottom-up models that combine the
quadratic superpotential adopted in Ref. DeWolfe:1999cp , and the confinement
mechanism in Ref. Witten:1998zw . We assume that the holographic description
of the dynamics be encapsulated in a model consisting of one real scalar,
coupled to gravity in $D=6$ dimensions, one of which is compacified on a
circle. Along the interesting, physical branch of solutions the circle shrinks
to a point at a finite position along the holographic dimension, intruducing a
dynamical scale, but the geometry is regular and smooth everywhere. What
results is the dual description of a putative family of four-dimensional,
confining theories, that at short distances are best described by the circle
compactification of a five-dimensional CFT in the presence of an operator with
non-trivial dimension given by ${\rm max}(\Delta,5-\Delta)$, where $\Delta$ is
a free parameter appearing in the scalar potential of the gravity theory.

The generic behaviour of the model, for each value of $\Delta$, is
qualitatively similar to that found in Refs. Elander:2020ial ; Elander:2020fmv
; Elander:2021wkc , and described above, as we shall show. The advantage of
the bottom-up approach—aside from having a much simpler bosonic action—is that
it allows us to vary $\Delta$. We are interested in the relation between
$\Delta$ and the mass spectrum, in particular for the lightest scalar, and in
proximity of the phase transition.

The paper is organised as follows. We start by defining the gravity theory in
$D=6$ dimensions, in Sec. II. We present several classes of classical
solutions in Sec. III, and then study the spectrum of small fluctuations for
one of these classes, the regular (confining) branch of solutions, in Sec. IV.
The free energy of the different classes of solutions is discussed in Sec. V,
where we analyse which branch of solutions is energetically favoured,
depending on the parameters of the model. We summarise the most salient
numerical results in Sec. VI, and outline future lines of research in Sec.
VII, while relegating to the Appendices many useful technical details.

##  II The model

This section introduces a model, built within the context of bottom-up
holography, that describes a real scalar field ${\phi}$ coupled to gravity in
$D=6$ dimensions, with a simple quartic scalar potential DeWolfe:1999cp . The
scalar field captures, in the gravity language, the effects of the deformation
of the dual five-dimensional CFT by a scalar operator, as well as the
formation of the corresponding condensate. Furthermore, one of the dimensions
is compactified on a circle (in both field theory as well as gravity). As we
shall see in Sec. III, this system admits solutions in which the circle
shrinks smoothly at the end of space in the IR, thus introducing a physical
low-energy scale, while mimicking (in analogy with top-down models) the effect
of confinement in what would be the dual, four-dimensional field theory
Witten:1998zw . We also find it convenient to provide the description of the
gravity theory in $D=5$ dimensions, obtained after reduction on the circle.

###  II.1 Action in six dimensions

We follow the conventions in Ref. Elander:2020csd  (see also references
therein), which we summarise in Appendix A. In order for the model to mimic
the dual of a four-dimensional confining theory, we choose to work in $D=6$
dimensions. The action is the following

$\displaystyle\mathcal{S}_{6}$
$\displaystyle=\mathcal{S}_{6}^{(bulk)}+\sum_{i=1,2}\mathcal{S}_{5,i}\,,$ (1)
$\displaystyle\mathcal{S}_{6}^{(bulk)}$
$\displaystyle=\int\mbox{d}^{6}x\sqrt{-\hat{g}_{6}}\,\bigg{\\{}\frac{\mathcal{R}_{6}}{4}-\frac{1}{2}\hat{g}^{\hat{M}\hat{N}}\partial_{\hat{M}}{\phi}\partial_{\hat{N}}\phi-{\mathcal{V}_{6}(\phi)}\bigg{\\}}\,,$
(2) $\displaystyle\mathcal{S}_{5,i}$
$\displaystyle=(-)^{i}\int\mbox{d}^{5}x\sqrt{-\tilde{\hat{g}}}\,\bigg{\\{}\frac{\mathcal{K}}{2}+\lambda_{i}(\phi)+f_{i}\left(\tilde{\hat{g}}_{\hat{M}\hat{N}}\right)\bigg{\\}}\bigg{|}_{\rho=\rho_{i}}\,,$
(3)

and besides the bulk part, $\mathcal{S}_{6}^{(bulk)}$, it contains also two
boundary actions, $\mathcal{S}_{5,i}$, localised at the boundaries of the
radial coordinate $\rho_{1}<\rho<\rho_{2}$. The space-time index is
$\hat{M}=0,1,2,3,5,6$. The extrinsic curvature $\mathcal{K}$, appearing in the
Gibbons-Hawking-York (GHY) term of the boundary actions, depends on the
induced metric on the boundaries, denoted $\tilde{\hat{g}}_{\hat{M}\hat{N}}$.

For the bulk potential $\mathcal{V}_{6}$, we choose to write

$\displaystyle\mathcal{V}_{6}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\partial{\mathcal{W}}_{6}(\phi)}{\partial{\phi}}\right)^{2}-\frac{5}{4}{\mathcal{W}}_{6}(\phi)^{2}$
(4) $\displaystyle=$
$\displaystyle-5-\frac{\Delta(5-\Delta)}{2}{\phi}^{2}-\frac{5\Delta^{2}}{16}{\phi}^{4}\,,$

where the superpotential is given by DeWolfe:1999cp

$\displaystyle{\mathcal{W}}_{6}(\phi)$ $\displaystyle\equiv$
$\displaystyle-2-\frac{\Delta}{2}{\phi}^{2}\,.$ (5)

The choice of a simple quadratic superpotential provides a neat field-theory
interpretation for backgrounds in which $\phi$ is non-zero: asymptotically in
the UV, the dual field theory flows towards a CFT in $D-1=5$ dimensions,
deformed by the insertion of an operator ${\cal O}$ with scaling dimension
given by ${\rm max}(\Delta,5-\Delta)$, and the two parameters appearing in the
solution of the corresponding second-order classical equations correspond in
field-theory terms to the coupling and condensate associated with ${\cal O}$.

We treat $\Delta$ as a free parameter—in top-down models, by contrast, its
counterpart descends from first principles. In all three examples discussed in
Refs. Elander:2020ial ; Elander:2020fmv ; Elander:2021wkc , the first-order
equations involve choices of $\Delta>(D-1)/2$ (where $D$ is equal to $6$, $7$,
and $5$, respectively, in those three models). Later in the paper, we will
discuss the differences emerging for $0<\Delta<(D-1)/2=5/2$. For the time
being, it suffices to notice that the counter-term used for holographic
renormalisation coincides with ${\mathcal{W}}_{6}$ when $\Delta<5/2$, while
for $\Delta>5/2$ one needs the associated superpotential

$\displaystyle\overline{\mathcal{W}}_{6}$ $\displaystyle=$
$\displaystyle-2-\frac{1}{2}(5-\Delta){\phi}^{2}-\frac{25(2\Delta-5)}{16(4\Delta-15)}{\phi}^{4}-\frac{125(2\Delta-5)\left(4\Delta^{2}-15\Delta+25\right)}{64(4\Delta-15)^{2}(6\Delta-25)}{\phi}^{6}\,+\,\cdots\,,$
(6)

which is only known perturbatively in $\phi$, yet also solves Eq. (4). In
Appendix B, we show this superpotential expanded to higher orders.
Pathological values of $\Delta=\frac{15}{4}\,,\,\frac{25}{6}\,,\cdots\,,$
appear at increasing orders in $\phi$. In the following, we will avoid using
these special pathological values. Another special case is $\Delta=5/2$, which
requires a separate treatment, as we shall see.

The boundary terms in Eq. (3) play an important role in this paper. They
determine the boundary conditions for the classical background solutions, via
the consistency requirements on the variational principle. They are used in
the calculation of the spectrum of fluctuations around such background
solutions. And finally, they enter into the calculation of the free energy—but
notice that this last calculation requires a modification of the UV-boundary
terms, in which $\lambda_{2}$ (and $f_{2}$) must be replaced by the counter-
terms required for holographic renormalisation.

###  II.2 Dimensional reduction to five dimensions

One of the dimensions is compact, the coordinate $0\leq\eta<2\pi$ describing a
circle. In reducing the action to five dimensions, we write the metric as

$\displaystyle\mbox{d}s_{6}^{2}$
$\displaystyle=e^{-2\chi}\mbox{d}x_{5}^{2}+e^{6\chi}\left(\mbox{d}\eta+\chi_{M}\mbox{d}x^{M}\right)^{2}\,,$
(7)

where the space-time index is $M=0,1,2,3,5$. The reduced action then becomes

$\displaystyle\mathcal{S}_{5}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{5}^{(bulk)}+\sum_{i=1,2}\mathcal{S}_{4,i}\,,$ (8)
$\displaystyle\mathcal{S}_{5}^{(bulk)}$ $\displaystyle=$
$\displaystyle\int\mbox{d}^{5}x\sqrt{-g_{5}}\,\left\\{\rule{-2.84544pt}{21.33955pt}\right.\frac{R}{4}-\frac{1}{2}g^{MN}\left[6\partial_{M}\chi\partial_{N}\chi\frac{}{}+\frac{}{}\partial_{M}{\phi}\partial_{N}{\phi}\right]-e^{-2\chi}\mathcal{V}_{6}(|\phi|)$
(9) $\displaystyle\hskip
68.28644pt-\frac{1}{16}e^{8\chi}g^{MP}g^{NQ}F^{(\chi)}_{MN}F^{(\chi)}_{PQ}\left.\rule{-2.84544pt}{21.33955pt}\right\\}\,,$
$\displaystyle\mathcal{S}_{4,i}$ $\displaystyle=$
$\displaystyle(-)^{i}\int\mbox{d}^{4}x\sqrt{-\tilde{g}}\,\bigg{\\{}\frac{K}{2}+e^{-\chi}\lambda_{i}(\phi)+e^{-\chi}f_{i}(\chi)\bigg{\\}}\bigg{|}_{\rho=\rho_{i}}\,,$
(10)

where the five-dimensional metric $g_{MN}$ has determinant $g_{5}$, the
induced metric on the boundaries is $\tilde{g}_{MN}$, the five-dimensional
Ricci scalar is $R$, and $K$ is the extrinsic curvature. The field strength
for the vector $\chi_{M}$ is given by
$F^{(\chi)}_{MN}=\partial_{M}\chi_{N}-\partial_{N}\chi_{M}$. The functions
$f_{i}$ of the six-dimensional theory depend explicitly on $\chi$, as is
required in order to obtain solutions that lift to geometries in six
dimensions in which the circle shrinks smoothly.

The scalars $\Phi^{a}=\\{\phi,\chi\\}$ describe a sigma-model coupled to
gravity, the action of which is the same as in Eq. (59) with $D=5$ and sigma-
model metric $G_{ab}={\rm diag}(1,\,6)$. We consider background solutions in
which $\chi_{M}=0$, while the metric $g_{MN}$, $\phi$, and $\chi$ depend on
the radial coordinate only. The metric in five dimensions takes the domain-
wall (DW) form

$\mbox{d}s_{5}^{2}=\mbox{d}r^{2}+e^{2A(r)}\mbox{d}x_{1,3}^{2}=e^{2\chi(\rho)}\mbox{d}\rho^{2}+e^{2A(\rho)}\mbox{d}x_{1,3}^{2}\,,$
(11)

where we have adopted the convenient choice of radial coordinate
$\mbox{d}\rho=e^{-\chi}\mbox{d}r$. The background fields satisfy the equations
of motion

$\displaystyle\partial_{\rho}^{2}\phi+(4\partial_{\rho}A-\partial_{\rho}\chi)\partial_{\rho}\phi$
$\displaystyle=\frac{\partial\mathcal{V}_{6}}{\partial{\phi}}\,,$ (12)
$\displaystyle\partial_{\rho}^{2}\chi+(4\partial_{\rho}A-\partial_{\rho}\chi)\partial_{\rho}\chi$
$\displaystyle=-\frac{\mathcal{V}_{6}}{3}\,,$ (13) $\displaystyle
3(\partial_{\rho}A)^{2}-\frac{1}{2}\left(\partial_{\rho}\phi\right)^{2}-3(\partial_{\rho}\chi)^{2}$
$\displaystyle=-\mathcal{V}_{6}\,,$ (14)

with boundary conditions given by

$\displaystyle\left(\partial_{\rho}\phi-\frac{\partial\lambda_{i}}{\partial\phi}\right)\bigg{|}_{\rho_{i}}$
$\displaystyle=0\,,\qquad\left(6\partial_{\rho}\chi+\lambda_{i}+f_{i}-\frac{\partial
f_{i}}{\partial\chi}\right)\bigg{|}_{\rho_{i}}=0\,,\qquad\left(\frac{3}{2}\partial_{\rho}A+\lambda_{i}+f_{i}\right)\bigg{|}_{\rho_{i}}=0\,.$
(15)

For vanishing $f_{i}=0$, one obtains solutions that lift to domain walls in
six dimensions.

The solutions satisfy the following equation:

$0=12(\partial_{\rho}{A})^{2}\,+\,{3}\partial_{\rho}^{2}{A}\,-\,3\partial_{\rho}\chi\partial_{\rho}A\,+\,4{\mathcal{V}}_{6}\,,$
(16)

which can be combined with Eq. (13) to yield the conservation law (see also
Ref. Elander:2020ial )

$\partial_{\rho}\left[\frac{}{}e^{4A-\chi}\left(\partial_{\rho}A-4\partial_{\rho}\chi\right)\right]=0\,.$
(17)

This defines a conserved quantity, which vanishes for DW solutions in six
dimensions, for which the metric $\mbox{d}s_{6}^{2}$ in Eq. (7) has the warp
factor defined by ${\cal A}\equiv A-\chi\,=\,3\chi$ (or $A=4\chi$).

##  III Classes of solutions

Here, we introduce the three main classes of solutions that we study in the
following. We explain the naming choices, before we exhibit the functional
form of the solutions, because they introduce some mild abuse of language,
driven mostly by analogy, which we want to alert the reader to. We start from
solutions to the first-order equations derived from the superpotential
formalism, along the lines of fake supergravity Freedman:2003ax . We call them
supersymmetric solutions, despite the fact that there is no supersymmetry—the
theory is bosonic. The reason for this naming convention is an analogy with
top-down models derived from higher-dimensional supergravity, for which the
first-order equations coincide with the Bogomol’nyi–Prasad–Sommerfield (BPS)
constraints.

The most important class of solutions of interest, and for which we compute
the spectrum of fluctuations, later in the paper, is denoted as confining
solutions, again with abuse of language. In models in which a lift to a higher
dimensional supergravity derived from a string theory exists, it might be
possible to compute the Wilson loop, along the lines of other holographic
models Rey:1998ik ; Maldacena:1998im ; Kinar:1998vq ; Brandhuber:1999jr ;
Avramis:2006nv ; Nunez:2009da , to expose the area-law expected from
confinement. This being a bottom-up model, such calculation cannot be carried
out, yet the solutions are completely smooth and introduce a mass gap in the
spectrum of fluctuations, which are weaker requirements for the gravity dual
of a confining theory.

Finally, we also introduce a set of singular backgrounds, that we call domain-
wall solutions, because in six dimensions they take the form of Poincaré
domain walls. There is again abuse of language: strictly speaking the
singularity signifies that they should not be taken literally as background
solutions, and used with caution—but see Ref. Gubser:2000nd . Our reason for
considering them is that, as we shall see, they teach us something important
about the other classes of solutions and their stability.

###  III.1 UV expansions

All the solutions we work with have the same asymptotic behaviour for large
$\rho$: they approach the ${\phi}=0$ critical point of ${\cal V}_{6}$, with
$\chi\simeq\frac{1}{3}\rho$ and $A\simeq\frac{4}{3}\rho$ (equivalently,
${\mathcal{A}}\simeq\rho$). Asymptotically in the UV, the gravity duals have
common interpretations in terms of (relevant or marginal) deformations of the
same five-dimensional CFT. We hence classify all the solutions in terms of a
power expansion in the small parameter $z\equiv e^{-\rho}$. The expansion
depends on five free parameters.

Two parameters are additive contributions to $\chi$ and $A$, which we denote
$\chi_{U}$ and $A_{U}$, respectively. One free parameter appears in $\chi$ in
the coefficient of the $z^{5}$ term, and we denote it as $\chi_{5}$. We fix an
additive contribution to $\chi_{5}$ so that if $A=4\chi$ (as is the case for
the DW solutions), then $\chi_{5}=0$. For generic values of $\Delta$, the
remaining two free parameters appear in the expansion of $\phi$ at orders
$z^{\Delta}$ and $z^{5-\Delta}$, and we call $\phi_{J}$ ($\phi_{V}$) the
coefficient in front of the term with the smallest (largest) exponent
$\Delta_{J}$ ($\Delta_{V}$). The expansions take the generic form:

$\displaystyle\phi(z)$
$\displaystyle=\phi_{J}z^{\Delta_{J}}\,+\,\cdots\,+\phi_{V}z^{\Delta_{V}}\,+\,\cdots\,,$
(18) $\displaystyle\chi(z)$
$\displaystyle=\chi_{U}-\frac{1}{3}\log(z)\,+\,\cdots+({\chi_{5}}+\cdots)z^{5}\,+\,\cdots\,,$
(19) $\displaystyle A(z)$
$\displaystyle=A_{U}-\frac{4}{3}\log(z)\,+\,\cdots\,.$ (20)

In the special case with $\Delta=5/2$, for which the two independent
parameters appear in front of the $z^{5/2}$ and $z^{5/2}\log(z)$ terms in the
expansion of $\phi$, we denote the former as $\phi_{V}$, and the latter as
$\phi_{J}$. As we shall see, these coefficients are very important in
determining the physical properties of the solutions and their field-theory
duals. For example, the free energy depends on a combination of
$\phi_{J}\phi_{V}$ and $\chi_{5}$. We further note that, without loss of
generality, one may restrict attention to solutions for which
$A_{U}=0=\chi_{U}$.

The expansion depends non-trivially on $\Delta$. For definiteness, we report
here the case $\Delta=3$, while more examples can be found in Appendix C:

$\displaystyle\phi(z)$
$\displaystyle=\phi_{J}z^{2}+\phi_{V}z^{3}-\frac{25}{48}\phi_{J}^{3}z^{6}-\frac{57}{80}\phi_{J}^{2}\phi_{V}z^{7}+\mathcal{O}\left(z^{8}\right)\,,$
(21) $\displaystyle\chi(z)$
$\displaystyle=\chi_{U}-\frac{1}{3}\log(z)-\frac{1}{24}\phi_{J}^{2}z^{4}+\left(\chi_{5}-\frac{2}{25}\phi_{J}\phi_{V}\right)z^{5}-\frac{1}{24}\phi_{V}^{2}z^{6}+\mathcal{O}\left(z^{8}\right)\,,$
(22) $\displaystyle A(z)$
$\displaystyle=A_{U}-\frac{4}{3}\log(z)-\frac{1}{6}\phi_{J}^{2}z^{4}+\left(\frac{1}{4}\chi_{5}-\frac{8}{25}\phi_{J}\phi_{V}\right)z^{5}-\frac{1}{6}\phi_{V}^{2}z^{6}+\mathcal{O}\left(z^{8}\right)\,.$
(23)

###  III.2 Supersymmetric solutions

As the scalar potential $\mathcal{V}_{6}$ is written in terms of a
superpotential ${\mathcal{W}}_{6}$, a special class of six-dimensional DW
solutions (for which $A(\rho)=4\chi(\rho)=\frac{4}{3}\mathcal{A}(\rho)$) is
recovered by solving the first-order equations

$\displaystyle\partial_{\rho}{\cal A}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{{\cal W}_{6}}\,=\,1+\frac{\Delta}{4}\phi^{2}\,,$
(24) $\displaystyle\partial_{\rho}\phi$ $\displaystyle=$
$\displaystyle\frac{\partial{{\cal W}_{6}}}{\partial\phi}\,=\,-\Delta\phi\,,$
(25)

which, as anticipated, we call supersymmetric solutions:

$\displaystyle\phi(\rho)$ $\displaystyle=$
$\displaystyle\phi_{c}\,e^{-\Delta\rho}\,=\,\phi_{c}\,z^{\Delta}\,,$ (26)
$\displaystyle\mathcal{A}(\rho)$ $\displaystyle=$
$\displaystyle\rho-\frac{1}{8}\phi_{c}^{2}\,e^{-2\Delta\rho}\,=\,-\log(z)-\frac{1}{8}\phi_{c}^{2}\,z^{2\Delta}\,.$
(27)

Besides $\phi_{c}$, a second, additive integration constant has been omitted
from ${\mathcal{A}}$. We recover the AdS6 geometry with $\phi_{c}=0$.

###  III.3 Confining solutions

The aforementioned confining solutions are such that the circle parametrised
by $\eta$ shrinks to zero size at some point $\rho_{o}$ of the radial
direction ${\rho}$—which is hence bounded from below as
$\rho_{o}<\rho_{1}\leq\rho<{\rho}_{2}\rightarrow+\infty$—and are also
completely regular and smooth, as the metric in six dimensions has finite
curvature invariants and there is no conical singularity. By power expanding
near the end of space, for small $(\rho-\rho_{o})$, we find that such
solutions have the following form

$\displaystyle\phi(\rho)$
$\displaystyle=\phi_{I}-\frac{1}{16}\Delta\phi_{I}\left(20+\Delta\left(5\phi_{I}^{2}-4\right)\right)(\rho-\rho_{o})^{2}+\mathcal{O}\left((\rho-\rho_{o})^{4}\right)\,,$
(28) $\displaystyle\chi(\rho)$
$\displaystyle=\chi_{I}+\frac{1}{3}\log(\rho-\rho_{o})+\frac{1}{288}\left(-80+8\left(\Delta-5\right)\Delta\phi_{I}^{2}-5\Delta^{2}\phi_{I}^{4}\right)(\rho-\rho_{o})^{2}+O\left((\rho-\rho_{o})^{4}\right)\,,$
(29) $\displaystyle A(\rho)$
$\displaystyle=A_{I}+\frac{1}{3}\log(\rho-\rho_{o})+\frac{7}{576}\left(80+\Delta\phi_{I}^{2}\left(40+\Delta\left(5\phi_{I}^{2}-8\right)\right)\right)(\rho-\rho_{o})^{2}+\mathcal{O}\left((\rho-\rho_{o})^{4}\right)\,,$
(30)

where $\phi_{I}$, $\chi_{I}$, $A_{I}$, and $\rho_{o}$ are integration
constants.

The induced metric on the $(\rho,\eta)$ sub-manifold is

$\displaystyle\mbox{d}s^{2}_{2}$ $\displaystyle=$
$\displaystyle\mbox{d}\rho^{2}\,+\,e^{6\chi}\mbox{d}\eta^{2}\,\simeq\,\mbox{d}\rho^{2}\,+\,e^{6\chi_{I}}(\rho-\rho_{o})^{2}\mbox{d}\eta^{2}\,+\,\cdots\,,$
(31)

which is the metric of the plane, provided $\eta$ has periodicity $2\pi$ and
we fix $\chi_{I}=0$, to avoid a conical singularity.

As a side remark, when $\Delta=0$, we can write the solution in closed form:

$\displaystyle\phi(\rho)$ $\displaystyle=$ $\displaystyle\phi_{I}\,,$ (32)
$\displaystyle\chi(\rho)$ $\displaystyle=$
$\displaystyle\chi_{o}-\frac{1}{5}\log\left[\cosh\left(\frac{5}{2}({\rho}-{\rho}_{o})\right)\right]+\frac{1}{3}\log\left[\sinh\left(\frac{5}{2}({\rho}-{\rho}_{o})\right)\right]\,,$
(33) $\displaystyle A(\rho)$ $\displaystyle=$
$\displaystyle{A_{o}+\frac{4}{15}\log\left[\sinh\left(5({\rho}-{\rho}_{o})\right)\right]+\frac{1}{15}\log\left[\tanh\left(\frac{5}{2}({\rho}-{\rho}_{o})\right)\right]}\,,$
(34)

where $\chi_{o}$ and $A_{o}$ are additive integration constants.

The curvature invariants in six dimensions—the Ricci scalar
$\mathcal{R}\equiv{\mathcal{R}}_{6}$, the Ricci tensor squared
$\mathcal{R}^{2}_{2}\equiv{\mathcal{R}_{6}}_{\hat{M}\hat{N}}{\mathcal{R}_{6}}^{\hat{M}\hat{N}}$,
and the Riemann tensor squared
$\mathcal{R}^{2}_{4}\equiv{\mathcal{R}_{6}}_{\hat{M}\hat{N}\hat{R}\hat{S}}{\mathcal{R}_{6}}^{\hat{M}\hat{N}\hat{R}\hat{S}}$—can
be written in terms of the non-trivial quantity

$\displaystyle d$ $\displaystyle\equiv$ $\displaystyle A-4\chi\,,$ (35)

which vanishes on the DW solutions. After using the equations of motion, we
find the following Elander:2020ial :

$\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle
6\mathcal{V}_{6}+2\big{(}\partial_{\rho}{\phi}\big{)}^{2}\,,$ (36)
$\displaystyle\mathcal{R}^{2}_{2}$ $\displaystyle=$ $\displaystyle
6\mathcal{V}_{6}^{2}+4\mathcal{V}_{6}\big{(}\partial_{\rho}{\phi}\big{)}^{2}+4\big{(}\partial_{\rho}{\phi}\big{)}^{4}\,,$
(37) $\displaystyle\mathcal{R}^{2}_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{250}\Bigg{(}32(\partial_{\rho}d)^{2}\Big{(}4\partial_{\rho}d\sqrt{36(\partial_{\rho}d)^{2}+15\sqrt{5}\sqrt{6\mathcal{R}^{2}_{2}-\mathcal{R}^{2}}-30\mathcal{R}}+24(\partial_{\rho}d)^{2}$
(38) $\displaystyle\hskip
85.35826pt+5\sqrt{5}\sqrt{6\mathcal{R}^{2}_{2}-\mathcal{R}^{2}}-10\mathcal{R}\Big{)}-25\left(\mathcal{R}^{2}-10\mathcal{R}^{2}_{2}\right)\Bigg{)}\,.$

All of these quantities are finite when $\rho\rightarrow\rho_{o}$, as can be
seen by explicitly using the IR expansions:

$\displaystyle\lim_{{\rho}\rightarrow{\rho}_{o}}\mathcal{R}$ $\displaystyle=$
$\displaystyle-30-{3\Delta(5-\Delta)}{\phi_{I}}^{2}-\frac{15\Delta^{2}}{8}{\phi_{I}}^{4}\,,$
(39) $\displaystyle\lim_{{\rho}\rightarrow{\rho}_{o}}\mathcal{R}^{2}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left(\lim_{{\rho}\rightarrow{\rho}_{o}}\mathcal{R}\right)^{2}\,,$
(40) $\displaystyle\lim_{{\rho}\rightarrow{\rho}_{o}}\mathcal{R}^{2}_{4}$
$\displaystyle=$
$\displaystyle\frac{1}{3}\left(\lim_{{\rho}\rightarrow{\rho}_{o}}\mathcal{R}\right)^{2}\,.$
(41)

Interestingly though, as long as $\Delta\neq 0$, all these invariants diverge
for $\phi_{I}\rightarrow\infty$, suggesting a priori that we should not be
allowed to take the value of $\phi_{I}$ to be arbitrarily large.

###  III.4 Singular domain-wall solutions

The last class of solutions of interest to this paper is given by singular DW
solutions. They obey the DW ansatz $A=4\chi=\frac{4}{3}{\cal A}$. Their IR
expansion depends explicitly on $\Delta$, and reads as follows:

$\displaystyle\phi(\rho)$ $\displaystyle=$
$\displaystyle\phi_{I}-\sqrt{\frac{2}{5}}\log(\rho-{\rho}_{o})+\frac{(\rho-{\rho}_{o})^{2}}{25920}\left(2\sqrt{10}(6\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})(\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})+2)\right.$
$\displaystyle\left.-23\Delta+60)+37\Delta+60)-\Delta(47\Delta+660)+5400)\right.$
$\displaystyle\left.+15\Delta\phi_{I}\left(-60\phi_{I}^{2}(6\Delta\log(\rho-{\rho}_{o})+\Delta)+6\sqrt{10}\phi_{I}(6\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})+1)\right.\right.$
$\displaystyle\left.\left.-23\Delta+60)-4(6\log(\rho-{\rho}_{o})(3\Delta\log(\rho-{\rho}_{o})(2\log(\rho-{\rho}_{o})+1)\right.\right.$
$\displaystyle\left.\left.-23\Delta+60)+37\Delta+60)+45\sqrt{10}\Delta\phi_{I}^{3}\right)\right)\,+\,{\cal
O}(({\rho}-{\rho}_{o})^{4})\,,$ $\displaystyle{\cal A}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{5}\log({\rho}-{\rho}_{o})+\frac{(\rho-{\rho}_{o})^{2}}{12960}\left(2(6\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})(\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})-10)\right.$
$\displaystyle\left.+7\Delta+60)-5(\Delta+60))+(1140-17\Delta)\Delta+5400)\right.$
$\displaystyle\left.+3\Delta\phi_{I}\left(2\sqrt{10}(5(\Delta+60)-6\log(\rho-{\rho}_{o})(3\Delta\log(\rho-{\rho}_{o})(2\log(\rho-{\rho}_{o})-5)+7\Delta+60))\right.\right.$
$\displaystyle\left.\left.+15\phi_{I}\left(2(6\Delta\log(\rho-{\rho}_{o})(3\log(\rho-{\rho}_{o})-5)+7\Delta+60)\right.\right.\right.$
$\displaystyle\left.\left.\left.+\Delta\phi_{I}\left(2\sqrt{10}(5-6\log(\rho-{\rho}_{o}))+15\phi_{I}\right)\right)\right)\right)\,+\,{\cal
O}(({\rho}-{\rho}_{o})^{4})\,.$

In these expressions, $\phi_{I}$ is an integration constant, and without loss
of generality we omitted an additive integration constant in $\mathcal{A}$. We
recall that the system of equations is symmetric under the change
$\phi\rightarrow-\phi$, hence a second branch of solutions can be obtained by
just changing the sign of $\phi$.

These solutions are singular at the end of space. Their nature in the gravity
theory is unclear, and they do not admit a transparent field theory
interpretation. In particular, we do not compute the spectrum of their
fluctuations—otherwise interpreted in terms of bound states in a putative dual
theory—as there is no clear sense in which this would provide information
about observable quantities. Yet, we do compute their free energy later in the
paper, because, as we shall see, for some choices of parameters, solutions of
this class are energetically favoured over the confining ones. We will come
back to this discussion in due time; we only anticipate here that this finding
will force us to restrict our physics discussions of the confining solutions
to a subregion of the full parameter space.

##  IV Mass spectrum of fluctuations

In this section, we restrict our attention to the confining solutions only.
Given a background solution in the gravity theory, one can linearise the
equations of motion of small fluctuations around it. The resulting mass
spectra of fluctuations of the bosonic fields correspond, following the gauge-
gravity duality dictionary, to spin-$0,1,2$ composite particles in the
(putative) dual confining field theory in four dimensions. To perform the
calculations, we adopt the gauge-invariant formalism developed in Refs.
Bianchi:2003ug ; Berg:2005pd ; Berg:2006xy ; Elander:2009bm ; Elander:2010wd .
We start from the scalar fluctuations
$\mathfrak{a}^{a}=\mathfrak{a}^{a}(q,\rho)$, where $q^{\mu}$ is the four-
momentum, which obey the following equations (see also Appendix A.1):

$\displaystyle{\Big{[}\partial^{2}_{\rho}+(4\partial_{\rho}A-\partial_{\rho}\chi)\partial_{\rho}-e^{2\chi-2A}q^{2}\Big{]}}\mathfrak{a}^{a}-e^{2\chi}\mathcal{X}^{a}_{\
c}\mathfrak{a}^{c}$ $\displaystyle=$ $\displaystyle 0\,.$ (44)

Because the sigma-model metric is simply $G_{ab}={\rm diag}(1,\,6)$, in the
basis $\Phi^{a}=\\{\phi,\chi\\}$, the sigma-model connection vanishes, greatly
simplifying the equations of motion, as $\mathcal{X}^{a}_{\ c}$ reads as
follows:

$\displaystyle\mathcal{X}^{a}_{\ c}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial}{\partial\Phi^{c}}\bigg{(}G^{ab}\frac{\partial(e^{-2\chi}\mathcal{V}_{6})}{\partial\Phi^{b}}\bigg{)}\,+\frac{4}{3\partial_{\rho}A}\bigg{[}\partial_{\rho}\Phi^{a}\frac{\partial(e^{-2\chi}\mathcal{V}_{6})}{\partial\Phi^{c}}+G^{ab}\frac{\partial(e^{-2\chi}\mathcal{V}_{6})}{\partial\Phi^{b}}\partial_{\rho}\Phi^{d}G_{dc}\bigg{]}$
(45)
$\displaystyle+\frac{16(e^{-2\chi}\mathcal{V}_{6})}{9(\partial_{\rho}A)^{2}}\partial_{\rho}\Phi^{a}\partial_{\rho}\Phi^{b}G_{bc}\,.$

In all these expressions, the functions $A$, $\Phi^{a}=\\{\phi,\chi\\}$, and
${\mathcal{V}}_{6}$ are evaluated on the background.

A discrete spectrum can be found by imposing the following boundary
conditions:111The equivalent form of the boundary conditions given in Eq. (14)
of Ref. Elander:2014ola  is convenient in numerical implementations.

$\left.\frac{}{}e^{-2\chi}\partial_{\rho}\Phi^{c}\partial_{\rho}\Phi^{d}G_{db}\partial_{\rho}\mathfrak{a}^{b}\right|_{\rho_{i}}=\left.\left[\frac{}{}\frac{3\partial_{\rho}A}{2}e^{-2A}q^{2}\delta^{c}_{\
b}+\partial_{\rho}\Phi^{c}\bigg{(}\frac{4\mathcal{V}_{6}}{3\partial_{\rho}A}\partial_{\rho}\Phi^{d}G_{db}+\frac{\partial\mathcal{V}_{6}}{\partial\Phi^{b}}\bigg{)}\right]\mathfrak{a}^{b}\right|_{\rho_{i}}\,,$
(46)

Physical composite states in the dual theory have mass $M^{2}=-q^{2}$,
corresponding to choices of $q^{2}$ for which the boundary conditions at the
two boundaries can be satisfied simultaneously.

The equations of motion for the (transverse and traceless) tensor fluctuations
$\mathfrak{e}^{\mu}_{\ \nu}$ are the following Elander:2018aub :

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left[\frac{}{}\partial^{2}_{{\rho}}+(4\partial_{{\rho}}A-\partial_{{\rho}}\chi)\partial_{{\rho}}-e^{2\chi-2A}q^{2}\right]\mathfrak{e}^{\mu}_{\
\nu}\,,$ (47)

while for the vector $\chi_{M}$ one looks at the gauge-invariant transverse
polarisations, which obey the bulk equation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
P^{\mu\nu}\left[\frac{}{}\partial^{2}_{{\rho}}+(2\partial_{{\rho}}A+7\partial_{{\rho}}\chi)\partial_{{\rho}}-e^{2\chi-2A}q^{2}\right]\chi_{\nu}\,,$
(48)

where $P^{\mu\nu}\equiv\eta^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}$. To compute
the spectrum of masses, $M>0$, of the bound states in the dual confining
theory, one can impose Neumann boundary conditions on these two kinds of
fluctuations.

The physical results can be recovered by first computing the spectrum for
finite values of $\rho_{1}$ and $\rho_{2}$, and then by repeating the
calculations to extrapolate toward the limits
${\rho}_{1}\rightarrow{\rho}_{o}$ and ${\rho}_{2}\rightarrow+\infty$
Elander:2010wd ; Elander:2013jqa ; Elander:2018aub . Equivalent results can be
obtained with an alternative numerical strategy, which improves the
convergence in such limits by making use of the UV and IR expansions of the
fluctuations. One decomposes them in dominant and subdominant modes, notices
that the boundary conditions are equivalent to suppressing the dominant
fluctuations, and then matches the solutions to the expansions, evaluated at
finite ${\rho}_{1,2}$. This non-trivial process requires knowing the
appropriate expansions at high orders in the small parameters ($z$ in the UV,
and ${\rho}-{\rho}_{o}$ in the IR), but has the great advantage that one does
not require extending the background solutions of the non-linear equations of
motion to numerically challenging values at large (small) $\rho-\rho_{o}$.
This strategy was successful in the study of the Klebanov-Strassler system
Berg:2006xy  and its baryonic branch Elander:2017cle ; Elander:2017hyr , and
we adopt it here. We report IR expansions, and one example of UV expansions,
for the fluctuations in Appendix D.

As explained and exemplified in Ref. Elander:2020csd , as well as Appendix E,
the probe approximation is equivalent to ignoring the coupling of the states
to the trace of the energy-momentum tensor, and hence discarding the mixing in
the physical states with the dilaton field. This approximation fails to
reproduce the mass of states with substantial overlap with the dilatation
operator. When such states are light, they should hence be interpreted as
approximate dilatons, turning this approximation into a diagnostic tool for
the identification of approximate dilatons. We will perform the analysis of
the mass spectrum in the probe approximation only for the choice of
$\Delta=5/2$, which turns out to be the most interesting in the context of
this paper.

Figure 1: Mass spectrum $\frac{M^{2}}{|M|}$ of fluctuations, computed for
confining backgrounds, with various choices of $\Delta$, as a function of the
IR parameter $\phi_{I}$. For each $\Delta$, we show the spectrum of spin-0
(blue), spin-1 (black), and spin-2 (red) states. The values of the IR and UV
cutoffs in the calculations are respectively given by
$\rho_{1}-\rho_{o}=10^{-9}$ and $\rho_{2}-\rho_{o}=5$. All masses are
normalised to the mass of the lightest spin-2 state. The vertical dashed lines
denote the critical value $\phi_{I}(c)$. Figure 2: Mass spectrum
$\frac{M^{2}}{|M|}$ of fluctuations, computed for confining backgrounds, with
various choices of $\Delta$, as a function of the IR parameter $\phi_{I}$. For
each $\Delta$, we show the spectrum of spin-0 (blue), spin-1 (black), and
spin-2 (red) states. The values of the IR and UV cutoffs in the calculations
are respectively given by $\rho_{1}-\rho_{o}=10^{-9}$ and
$\rho_{2}-\rho_{o}=5$. All masses are normalised to the mass of the lightest
spin-2 state. The vertical dashed lines denote the critical value
$\phi_{I}(c)$. Figure 3: Mass spectrum $\frac{M^{2}}{|M|}$ of fluctuations,
computed for confining backgrounds, with various choices of $\Delta$, as a
function of the IR parameter $\phi_{I}$. For each $\Delta$, we show the
spectrum of spin-0 (blue), spin-1 (black), and spin-2 (red) states. The values
of the IR and UV cutoffs in the calculations are respectively given by
$\rho_{1}-\rho_{o}=10^{-9}$ and $\rho_{2}-\rho_{o}=5$. All masses are
normalised to the mass of the lightest spin-2 state. The vertical dashed lines
denote the critical value $\phi_{I}(c)$. Figure 4: Mass spectrum
$\frac{M^{2}}{|M|}$ of fluctuations, computed for confining backgrounds, with
various choices of $\Delta$, as a function of the IR parameter $\phi_{I}$. For
each $\Delta$, we show the spectrum of spin-0 (blue), spin-1 (black), and
spin-2 (red) states. The values of the IR and UV cutoffs in the calculations
are respectively given by $\rho_{1}-\rho_{o}=10^{-9}$ and
$\rho_{2}-\rho_{o}=5$. All masses are normalised to the mass of the lightest
spin-2 state. The vertical dashed lines denote the critical value
$\phi_{I}(c)$.

The mass spectra are computed numerically, and we report the results in Figs.
1, 2, 3, and 4, for a few significant choices of $\Delta$. For each of them,
we compute the spectrum of spin-0, spin-1, and spin-2 fluctuations. We
conveniently normalise the spectra so that the lightest tensor state has mass
$M=1$. For each value of $\Delta$, we check the convergence of the spectrum,
as a function of ${\rho}_{1}$ and ${\rho}_{2}$ independently, and report
numerical results obtained with choices for which the dependence on the two
cutoffs can be neglected in respect to our numerical accuracy goal of $0.5\%$.
Finally, we find it convenient to report in the plots also the critical value
of $\phi_{I}(c)$, obtained from the numerical study of the free energy, as
detailed in the next section.

##  V Free energy

The free energy density is given by the following expression:

$\displaystyle{\cal F}$ $\displaystyle=$
$\displaystyle-\lim_{\rho_{2}\rightarrow+\infty}e^{4A-\chi}\Big{(}\frac{3}{2}\partial_{\rho}A+\mathcal{W}_{2}\Big{)}\Big{|}_{\rho_{2}}\,,$
(49)

where ${\cal W}_{2}={\mathcal{W}}_{6}$ if $\Delta<5/2$, or ${\cal
W}_{2}=\overline{\mathcal{W}}_{6}$ if $\Delta>5/2$. This expression has been
obtained by adapting the results in Ref. Elander:2020ial ; it includes the
contribution of the bulk action, evaluated by imposing the equations of
motion, the contribution of the boundary terms in the action, and the
appropriate UV-localised counter-term ${\mathcal{W}}_{2}$, which is required
by the rules of holographic renormalisation Bianchi:2001kw ; Skenderis:2002wp
; Papadimitriou:2004ap , as it removes the UV divergences. The conservation
law in Eq. (17) has been used to evaluate at $\rho_{2}$ (the UV boundary) a
term naturally defined at $\rho_{1}$ (the IR boundary).

For each choice of the parameter $\Delta$, one expresses ${\cal F}$ in terms
of the coefficients appearing in the UV expansion of the background, which can
be extracted by matching the expansions to numerical solutions. Care must be
taken to obtain results that converge as $\rho_{2}\rightarrow+\infty$. We
empirically find that, for $\Delta<5/2$, the free energy density is

$\displaystyle{\cal F}$ $\displaystyle=$
$\displaystyle-\frac{1}{40}e^{4A_{U}-\chi_{U}}\left(16\Delta\left(\frac{5}{2}-\Delta\right)\phi_{J}\phi_{V}\frac{}{}-75\chi_{5}\right)\,,$
(50)

while for $\Delta>5/2$—and retaining in $\overline{\mathcal{W}}_{6}$ a
sufficient number of terms to ensure cancellation of all the divergences—the
free energy density is

$\displaystyle{\cal F}$ $\displaystyle=$
$\displaystyle-\frac{1}{40}e^{4A_{U}-\chi_{U}}\left(16(\Delta-5)\left(\frac{5}{2}-\Delta\right)\phi_{J}\phi_{V}\frac{}{}-75\chi_{5}\right)\,.$
(51)

We explicitly checked that this expression is accurate for all values of
$\Delta$ considered in this paper. In the special case in which the operator
has dimension $\Delta=5/2$, we find that

$\displaystyle{\cal W}_{2}$ $\displaystyle=$
$\displaystyle-2-\frac{5}{4}\phi^{2}\left(1+\frac{2}{5\log(k\,z)}\right)\,,$
(52)

and the free energy density is

$\displaystyle\mathcal{F}=\frac{1}{40}e^{4A_{U}-\chi_{U}}\bigg{(}20\phi_{J}\phi_{V}-4\phi_{J}^{2}+75\chi_{5}-20\phi_{J}^{2}\log(k)\bigg{)}\,.$
(53)

The appearance of a logarithm introduces a residual scheme dependence, encoded
in the parameter $k$. In the following, we make the choice $k=\Lambda$.

Following Refs. Elander:2020ial ; Elander:2020fmv ; Elander:2021wkc , we find
it convenient to define a scale $\Lambda$ as follows Csaki:2000cx :

$\Lambda^{-1}\equiv\int_{\rho_{o}}^{\infty}\mbox{d}\rho\,e^{\chi(\rho)-A(\rho)}\,,$
(54)

with $\rho_{o}$ the end of space—other choices are admissible, but this choice
makes contact with earlier studies, where it has been shown that it allows to
compare the confining and DW classes of solutions. We then express all of the
quantities of interest in units of such scale $\Lambda$, by defining the
rescaled free energy density

$\displaystyle\hat{\mathcal{F}}$ $\displaystyle\equiv$
$\displaystyle\frac{\mathcal{F}}{\Lambda^{5}}\,,$ (55)

and the rescaled source as

$\displaystyle\hat{\phi}_{J}$ $\displaystyle\equiv$
$\displaystyle\frac{\phi_{J}}{\Lambda^{\Delta_{J}}}\,,$ (56)

where $\Delta_{J}$ is the dimension of the deforming coefficient (source)
associated with the dual operator of dimension $\Delta_{V}\equiv
5-\Delta_{J}$. We hence define also the rescaled condensates to be

$\displaystyle\hat{\phi}_{V}$ $\displaystyle\equiv$
$\displaystyle\frac{\phi_{V}}{\Lambda^{\Delta_{V}}}\,,$ (57)
$\displaystyle\hat{\chi}_{5}$ $\displaystyle\equiv$
$\displaystyle\frac{\chi_{5}}{\Lambda^{5}}\,.$ (58)

Given a background solution, in order to compute the free energy we proceed as
follows. First, we match it to the UV expansions, and determine coefficients
$A_{U}$, $\chi_{U}$, $\phi_{J}$, $\phi_{V}$, and $\chi_{5}$. We then shift
additively both the radial coordinate ${\rho}$ and the definition of the
function $A({\rho})$, to impose the constraints $A_{U}=0=\chi_{U}$. We repeat
the determination of the coefficients, and enter the results in the expression
for the free energy density. We compute the scale $\Lambda$ for each
background—after the aforementioned shifts of ${\rho}$ and $A({\rho})$—and
finally plot the resulting, rescaled quantities. Examples are shown in Figs.
5, 6, 7, and 8. We do so for both confining and DW singular solutions, and
show them together for each $\Delta$.

Figure 5: The free energy density $\hat{\cal F}$ as a function of the source
$\hat{\phi}_{{J}}$, in units of the scale $\Lambda$, for various choices of
$\Delta$. The black curve denotes the confining solutions, while the red and
blue solutions are singular domain-wall ones. Figure 6: The free energy
density $\hat{\cal F}$ as a function of the source $\hat{\phi}_{{J}}$, in
units of the scale $\Lambda$, for various choices of $\Delta$. The black curve
denotes the confining solutions, while the red and blue solutions are singular
domain-wall ones. Figure 7: The free energy density $\hat{\cal F}$ as a
function of the source $\hat{\phi}_{{J}}$, in units of the scale $\Lambda$,
for various choices of $\Delta$. The black curve denotes the confining
solutions, while the red and blue solutions are singular domain-wall ones.
Figure 8: The free energy density $\hat{\cal F}$ as a function of the source
$\hat{\phi}_{{J}}$, in units of the scale $\Lambda$, for various choices of
$\Delta$. The black curve denotes the confining solutions, while the red and
blue solutions are singular domain-wall ones.

By inspection of the figures, we see the emergence of a common pattern; in all
cases, the confining solutions minimise the free energy, $\hat{\mathcal{F}}$,
for small choices of the source $|\hat{\phi}_{J}|$, while the singular, DW
solutions have the lowest $\hat{\cal F}$ for larger values of the source.
There is a critical value of $|\hat{\phi}_{J}(c)|$, corresponding to a
critical value of the IR parameter $|\phi_{I}(c)|$, where a first-order phase
transition occurs. For $|\phi_{I}|\leq|\phi_{I}(c)|$, the confining solutions
can be physically realised, while for $|\phi_{I}|>|\phi_{I}(c)|$, the physical
solution is unknown, so that we are forced to discard these latter regions of
parameter space. We are not claiming that the singular solutions are
physically realised, but simply that some other branch of solutions, not
identified in this study, might exist, and take over the long-distance
dynamics in these cases.

When inspected more closely, the figures show other interesting features. For
$\Delta\geq 5/2$, both confining and singular solutions cease to exist above
some value of $|\hat{\phi}_{J}({\rm max})|>|\hat{\phi}_{J}(c)|$. As we already
know (from the aforementioned arguments) that, at large
$|\hat{\phi}_{J}|>|\hat{\phi}_{J}(c)|$, another branch of solutions must exist
(possibly in a more complete theory), this is not a particularly problematic
finding. While this finding agrees with the results of the analysis of three
distinct examples of top-down holographic models, reported in Refs.
Elander:2020ial ; Elander:2020fmv ; Elander:2021wkc , it is interesting that
this feature is absent for $\Delta<5/2$, in which case confining and singular
solutions exist for any choices of $\hat{\phi}_{J}$.

In thermodynamics, one can use the concavity theorems to discuss the stability
of possible solutions, determined on the basis of the shape of the functional
dependence of the free energy on the control parameters. These arguments do
not transfer directly to the case in question, because of the presence of
divergences and scheme dependences (see also Ref. Bobev:2013cja , for examples
of similar considerations in a different context), and caution has to be
exercised. Yet, by comparing Figs. 5, 6, 7, and 8 with Figs. 1, 2, 3, and 4,
one notices the following.

  * •

A tachyon exists in the spectra of confining theories for values of
$\Delta\gtrsim 1.8$, but only in a portion of parameter space (equivalently,
for some range of $\phi_{I}$).

  * •

The portion of parameter space in which the tachyon appears is always past the
phase transition.

  * •

There is a region of parameter space over which the mass of the lightest
scalar is parameterically small, near the point at which it turns tachyonic.
But once more: this happens only past the phase transition, in regions in
which the confining solutions are, at best, metastable.

We performed an extensive study of the free energy for values of $\Delta\simeq
1.8$, nearby where the tachyon appears—see also the details in Appendix F. We
find that the free energy of the confining backgrounds is a monotonic function
for $\Delta$ below $\Delta\simeq 1.7$. As expected, the relation between the
behavior of ${\cal F}$ and the appearance a tachyon is not precise, and
furthermore these features always appear in a region of parameter space that
has already been excluded, on the basis of earlier considerations. We hence
report on this feature, for completeness, but do not explore it any further.

##  VI Summary

We collect in Table 1 our numerical results: for representative values of
$\Delta$, we tabulate the critical parameters $\hat{\phi}_{J}({\rm c})$
(source) and $\phi_{I}({\rm c})$ (IR asymptotic value of $\phi$), the mass of
the lightest scalar $M(c)$ (in units of the mass of the lightest tensor), in
proximity of the transition. For completeness, we also list the value of the
two parameters representing the condensates in the dual field theory:
$\hat{\phi}_{V}(c)$ and $\hat{\chi}_{5}(c)$. We show in Fig. 9 the mass
spectrum and free energy for the $\Delta=5/2$ case, and in Fig. 10 we display
the mass spectrum computed at the critical value $\phi_{I}(c)$, for each
choice of $\Delta$.

The main results of the paper can be summarised as follows. The model allows
us to study the spectrum and the free energy for any generic value of the
parameter $\Delta$, which is equal either to the dimensionality of the
coupling deforming the dual CFT, or to the associated condensate. Furthermore,
for each $\Delta$ we restrict attention to regular backgrounds, the dual field
theory of which has a mass gap and a discrete spectrum of bound states. These
solutions of the gravity equations of motion in five dimensions lift to
completely regular and smooth solutions in six dimensions, and admit an end of
space in the radial direction ${\rho}>{\rho}_{o}$. We compute the spectrum of
fluctuations of solutions of this type, which with abuse of notation we call
confining. We find that for all values of the parameter $0<\Delta<5$ that we
studied, there exists a first-order phase transition, that bounds from above
the source associated with the scalar field $\phi$. This is demonstrated by
the existence of domain-wall solutions in six dimensions that, for large
enough values of the deforming parameters, are energetically favoured (despite
being singular) over the confining ones. We find that for $\Delta\gtrsim 1.8$,
the mass spectrum contains a tachyon, provided the deforming parameter is
large enough. But this tachyon always appears only in an unphysical portion of
parameter space, well beyond the phase transition. Conversely, in the physical
portion of parameter space, all fluctuations have positive $M^{2}>0$, and are
never parametrically light.

By inspecting Figs. 1, 2, 3, and 4, we notice that the mass of the lightest
state in the spectrum, in units of the mass of the lightest tensor, becomes
smaller when the source increases towards its critical value. Hence, the
minimum value $M(c)$ of the mass of the lightest scalar is found in immediate
proximity of the transition. For each choice of $\Delta$, we compute the mass
spectrum of bound states of the dual theory precisely at the phase transition
point, which is displayed in Fig. 10. We find two interesting, unexpected
features, which we highlight in closing this section.

First, the mass of the lightest scalar state, has a minimum in proximity of
$\Delta=5/2$, in which case the mass of such state is approximately one third
of the mass of the tensor. We show explicitly in the top panel of Fig. 9 the
mass spectrum computed in the probe approximation, which shows significant
disagreement with the mass of the lightest scalar for large values of
$\phi_{I}$. In this case, the lightest scalar particle has a substantial
overlap with the dilaton Elander:2020csd . Unfortunately, this feature appears
only inside the metastable (or tachyonic) region(s) of parameter space, while
the probe approximation captures well the mass of the lightest scalar in the
the physical region, when $|\phi_{I}|\leq|\phi_{I}(c)|$.

Table 1: Summary table of our results. For each $\Delta$, we report the
critical value of the normalised source $\hat{\phi}_{J}({\rm c})$ at the phase
transition, the critical value of the IR expansion parameter $\phi_{I}({\rm
c})$, and the value of the mass $M({\rm c})$ of the lightest scalar state at
the transition—expressed in units of the mass of the lightest spin-2 state for
the same choices of parameters. For completeness, we also report the value of
the two parameters linked to the condensates in the dual field theory:
$\hat{\phi}_{V}(c)$ and $\hat{\chi}_{5}(c)$.
$~{}~{}~{}\Delta~{}~{}~{}$ | $~{}~{}~{}\hat{\phi}_{J}({\rm c})~{}~{}~{}$ | $~{}~{}~{}\phi_{I}({\rm c})~{}~{}~{}$ | $~{}~{}~{}M({\rm c})~{}~{}~{}$ | $~{}~{}~{}\hat{\phi}_{V}({\rm c})~{}~{}~{}$ | $~{}~{}~{}\hat{\chi}_{5}({\rm c})~{}~{}~{}$
---|---|---|---|---|---
$~{}~{}0.50~{}~{}$ | $2.373$ | $1.89$ | $0.571$ | $-0.063$ | $-0.176$
$~{}~{}1.00~{}~{}$ | $1.149$ | $0.845$ | $0.553$ | $-0.208$ | $-0.328$
$~{}~{}1.50~{}~{}$ | $0.990$ | $0.567$ | $0.512$ | $-0.369$ | $-0.368$
$~{}~{}1.70~{}~{}$ | $1.010$ | $0.494$ | $0.482$ | $-0.471$ | $-0.377$
$~{}~{}1.75~{}~{}$ | $1.022$ | $0.477$ | $0.473$ | $-0.504$ | $-0.380$
$~{}~{}1.80~{}~{}$ | $1.038$ | $0.459$ | $0.464$ | $-0.539$ | $-0.382$
$~{}~{}1.85~{}~{}$ | $1.059$ | $0.442$ | $0.455$ | $-0.579$ | $-0.384$
$~{}~{}1.90~{}~{}$ | $1.084$ | $0.424$ | $0.446$ | $-0.625$ | $-0.386$
$~{}~{}1.95~{}~{}$ | $1.115$ | $0.407$ | $0.436$ | $-0.677$ | $-0.388$
$~{}~{}2.00~{}~{}$ | $1.153$ | $0.388$ | $0.427$ | $-0.736$ | $-0.390$
$~{}~{}2.25~{}~{}$ | $1.567$ | $0.281$ | $0.385$ | $-1.276$ | $-0.401$
$~{}~{}2.30~{}~{}$ | $1.756$ | $0.254$ | $0.379$ | $-1.496$ | $-0.403$
$~{}~{}2.35~{}~{}$ | $2.045$ | $0.223$ | $0.374$ | $-1.821$ | $-0.405$
$~{}~{}2.40~{}~{}$ | $2.554$ | $0.186$ | $0.370$ | $-2.371$ | $-0.408$
$~{}~{}2.45~{}~{}$ | $3.748$ | $0.136$ | $0.367$ | $-3.618$ | $-0.410$
$~{}~{}2.49~{}~{}$ | $8.750$ | $0.063$ | $0.368$ | $-8.691$ | $-0.412$
$~{}~{}2.499~{}~{}$ | $27.82$ | $0.020$ | $0.369$ | $-27.80$ | $-0.413$
$~{}~{}2.50~{}~{}$ | $-0.295$ | $0.107$ | $0.359$ | $0.439$ | $-0.411$
$~{}~{}2.501~{}~{}$ | $147.3$ | $0.107$ | $0.359$ | $-147.2$ | $-0.411$
$~{}~{}2.51~{}~{}$ | $14.79$ | $0.107$ | $0.362$ | $-14.69$ | $-0.411$
$~{}~{}2.55~{}~{}$ | $3.001$ | $0.108$ | $0.372$ | $-2.899$ | $-0.411$
$~{}~{}2.60~{}~{}$ | $1.528$ | $0.108$ | $0.384$ | $-1.426$ | $-0.411$
$~{}~{}2.65~{}~{}$ | $1.038$ | $0.108$ | $0.397$ | $-0.935$ | $-0.411$
$~{}~{}2.70~{}~{}$ | $0.793$ | $0.109$ | $0.409$ | $-0.689$ | $-0.411$
$~{}~{}2.75~{}~{}$ | $0.647$ | $0.109$ | $0.421$ | $-0.542$ | $-0.411$
$~{}~{}3.00~{}~{}$ | $0.358$ | $0.113$ | $0.481$ | $-0.246$ | $-0.411$
$~{}~{}3.50~{}~{}$ | $0.225$ | $0.127$ | $0.586$ | $-0.068$ | $-0.411$
$~{}~{}4.00~{}~{}$ | $0.189$ | $0.150$ | $0.616$ | $-0.102$ | $-0.410$
$~{}~{}4.80~{}~{}$ | $0.136$ | $0.204$ | $0.620$ | $-0.179$ | $-0.410$

Second, this summary plot displays a peculiar discontinuity at $\Delta=5/2$
(see the inset of Fig. 10). The numerical value for the mass of the lightest
scalar at $\Delta=5/2$ aligns well with those obtained for $\Delta>5/2$, and
is the absolute minimum of the mass. However, the sequence of masses of the
lightest scalar state one obtains for $\Delta<5/2$ converges to a slightly
higher value. We do not know why this second feature emerges, and highlight to
the reader the intrinsic numerical difficulty of some of the analysis we
carried out. Yet, this discontinuity is a small effect, when compared to the
much more significant first feature we found, namely that the mass of the
lightest scalar is minimised for $\Delta=5/2$.

Figure 9:  The mass spectrum of fluctuations for $\Delta=5/2$, displaying the
spin-0 (blue), spin-1 (black), and spin-2 (red) states, in units of the mass
of the lightest spin-2 state (left panel). The spectrum also includes the
masses of the scalars computed in the probe approximation (orange), for
comparison with the complete calculation (blue). The values of the IR and UV
cutoffs in the calculations are respectively given by
$\rho_{1}-\rho_{o}=10^{-9}$ and $\rho_{2}-\rho_{o}=5$. The (normalised) free
energy density $\hat{\cal F}$, as a function of the source $\hat{\phi}_{J}$,
for $\Delta=5/2$ (right panel). The black curve denotes the confining
solutions, while the red and blue solutions are singular domain-wall ones. The
vertical, dashed line in the left panel indicates the position of the first-
order phase transition evident in the right panel. Figure 10: The mass
spectrum of fluctuations of the gravity theory (corresponding to the bound
states of the dual field theory) evaluated at the critical $\phi_{I}(c)$ for
each choice of $\Delta$. We display the spin-0 (blue), spin-1 (black), and
spin-2 (red) states. The values of the IR and UV cutoffs in the calculations
are respectively given by $\rho_{1}-\rho_{o}=10^{-9}$ and
$\rho_{2}-\rho_{o}=5$. All masses are normalised to the mass of the lightest
spin-2 state.

##  VII Outlook

In a broad class of bottom-up holographic models, in which the background
geometry ends smoothly, so as to introduce a mass gap in the dual field theory
in a way that mimics a confining theory, and furthermore in the presence of an
additional deformation (and the corresponding condensate), we found evidence
of the presence of a first-order phase transition. The nature of the model
allows one to treat $\Delta$, the dimensionality of the
deformation/condensate, as a free parameter, and study the physics as a
function of $\Delta$. All the fluctuations of the gravity backgrounds have
positive mass squared, $M^{2}>0$, for all values considered, $0<\Delta<5$, in
the physical portion of the parameter space, before the appearance of the
phase transition. Interestingly, in portions of the parameter space well
beyond the phase transition, the lightest scalar state in the theory becomes
first an approximate dilaton, its mass being paramaterically suppressed, and
then a tachyon. All of these results confirm and generalise the findings in
the three top-down models studied in Refs. Elander:2020ial ; Elander:2020fmv ;
Elander:2021wkc , as well as the expectations of Refs. Gorbenko:2018ncu ;
Gorbenko:2018dtm ; Pomarol:2019aae .

Restricting attention to the physical region of parameter space, in which the
confining solutions are stable, we found some evidence that the mass of the
lightest state in the spectrum becomes smaller in the limit $\Delta\rightarrow
5/2$— half of the dimension of the space-time of the CFT that the dual theory
flows from in the UV—in line with the suggestion, in Ref. Kaplan:2009kr , that
such values of $\Delta$ are special. However, the lightest mass is light, but
not parametrically so, as predicted in Ref. Pomarol:2019aae .

This study suggests that the phase transition we uncovered in this class of
models cannot be rendered arbitrarily weak, nor of second order, and the
lightest state cannot be a dilaton with tunable, small mass. The phenomena
associated with confinement in the dual field theory, and its specific
implementation—which we borrowed from Witten’s model Witten:1998zw —are
playing an important role in yielding this conclusion. These findings support
the arguments according to which inferring the properties of long-distance
dynamics on the basis solely of the short-distance characterisation of the
theory (its fixed points, and the classification of their deformations), is
misleading, as the RG flow away from fixed points, ultimately leading to
confinement, adds important non-perturbative effects.

It is premature to try to generalise these conclusions: the class of models
analysed here does not cover all possible realisations of confining field
theories, and possibly the limitation of all these studies descends from the
fact that confinement is described always in the same way, in terms of a
shrinking circle in the internal geometry of a higher-dimensional gravity
theory. For instance, top-down holographic models in which the analysis of the
spectrum of fluctuations has been performed include Refs. Elander:2018gte ,
and Elander:2017cle ; Elander:2017hyr , and the results (in particular,
regarding the presence of light states) are model-dependent, as the mechanism
introducing a mass gap is different from the mechanism of interest here. We
remind the Reader that in Refs. Elander:2020fmv ; Elander:2021kxk  a similar
model, but derived from maximal supergravity in seven dimensions, was studied.
The main difference here is the fact that, in the bottom-up model considered
in the present paper, $\Delta$ is a free parameter. The search for models that
realise a weak phase transition, and in which a parametrically light dilaton
can be realised, in the presence of a nearby instability in the parameter
space of the theory, continues.

We conclude by anticipating that we will further extend this six-dimensional
model, along lines similar to those followed in Ref. Elander:2021kxk , in the
complementary context of composite Higgs models, in future publications SO(5)
; vacuum ; EWSB . In this different context, we will extend the sigma-model to
incorporate an internal gauge symmetry. We will study how to break the
corresponding global symmetry in the field theory, how to (weakly) gauge a
subgroup, and how to exploit vacuum (mis-)alignment to construct a composite
Higgs model, with semi-realistic features—in particular in reference to
electroweak symmetry breaking.

###### Acknowledgements.

We would like to thank Ed Bennett for assisting in the preparation of the data
release. The work of AF has been supported by the STFC Consolidated Grant No.
ST/V507143/1. The work of MP has been supported in part by the STFC
Consolidated Grants No. ST/P00055X/1 and No. ST/T000813/1. MP received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program under Grant Agreement No. 813942.
Research Data -  All data displayed in the figures of this manuscript can be
downloaded from Ref. ZEAP . We refer the reader to Refs. Bennett:2022klt ;
Athenodorou:2022ixd  and references therein for recent surveys on open
science. Open Access Statement -  For the purpose of open access, the authors
have applied a Creative Commons Attribution (CC BY) licence to any Author
Accepted Manuscript version arising.

##  Appendix A Sigma-model coupled to gravity

For the purpose of fixing our conventions, we write here the action of the
two-derivative sigma-model in $D$ dimensions, consisting of $n$ scalars
$\Phi^{a}$, with $a=1,\,\cdots,\,n$, coupled to gravity:

$\displaystyle\mathcal{S}$
$\displaystyle=\int\mbox{d}^{D}x\sqrt{-g}\,\bigg{[}\frac{R}{4}-\frac{1}{2}g^{MN}G_{ab}\partial_{M}\Phi^{a}\partial_{N}\Phi^{b}-{\mathcal{V}}(\Phi^{a})\bigg{]}\,.$
(59)

We denote by $M=0,\cdots,\,3,\,5,\,\cdots,\,D$ the $D$-dimensional space-time
indexes. The $D$-dimensional metric $g_{MN}$ has determinant $g$, and
signature mostly $+$. The $D$-dimensional Ricci scalar is denoted by $R$. The
sigma-model metric is $G_{ab}$, and its inverse is $G^{ab}$. The potential
$\mathcal{V}$ is a function of the scalars $\Phi^{a}$.

If we adopt the ansatz for the background solutions, that the metric is of the
DW form, and that the scalar fields only depend on the radial coordinate,

$\displaystyle\mbox{d}s_{D}^{2}$ $\displaystyle=$
$\displaystyle\mbox{d}r^{2}+e^{2{A}(r)}\mbox{d}x_{1,D-2}^{2}\,,$ (60)
$\displaystyle\Phi^{a}$ $\displaystyle=$ $\displaystyle\Phi^{a}(r)\,,$ (61)

then the equations of motion are given by

$\displaystyle\partial_{r}^{2}\Phi^{a}+(D-1)\partial_{r}A\partial_{r}\Phi^{a}+{\mathcal{G}}^{a}_{\,\,\,\,bc}\partial_{r}\Phi^{b}\partial_{r}\Phi^{c}-G^{ab}\frac{\partial\mathcal{V}}{\partial\Phi^{b}}$
$\displaystyle=$ $\displaystyle 0\,,$ (62)
$\displaystyle(D-1)\left(\partial_{r}A\right)^{2}+\partial_{r}^{2}A+\frac{4}{D-2}\mathcal{V}$
$\displaystyle=$ $\displaystyle 0\,,$ (63)
$\displaystyle(D-1)(D-2)\left(\partial_{r}A\right)^{2}-2G_{ab}\partial_{r}\Phi^{a}\partial_{r}\Phi^{b}+4\mathcal{V}$
$\displaystyle=$ $\displaystyle 0\,,$ (64)

where the sigma-model connection is

$\displaystyle{\cal G}^{d}_{\,\,\,\,ab}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}G^{dc}\left(\frac{}{}\partial_{a}G_{cb}+\partial_{b}G_{ca}-\partial_{c}G_{ab}\right)\,.$
(65)

Furthermore, if one finds a solution, $\mathcal{W}(\Phi^{a})$, to the
following partial differential equation:

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G^{ab}\frac{\partial\mathcal{W}}{\partial\Phi^{a}}\frac{\partial\mathcal{W}}{\partial\Phi^{b}}-\frac{D-1}{D-2}\mathcal{W}^{2}\,,$
(66)

then any solution to the first-order equations

$\displaystyle\partial_{r}A$ $\displaystyle=$
$\displaystyle-\frac{2}{D-2}\mathcal{W}\,,$ (67)
$\displaystyle\partial_{r}\Phi^{a}$ $\displaystyle=$ $\displaystyle
G^{ab}\frac{\partial\mathcal{W}}{\partial\Phi^{b}}\,,$ (68)

is also a solution of the second-order classical Eqs. (62), (63), and (64).

###  A.1 Linearised equations for scalar fluctuations

In general, the linearised equations of motion can be put in gauge-invariant
form following the formalism in Refs. Bianchi:2003ug ; Berg:2005pd ;
Berg:2006xy ; Elander:2009bm ; Elander:2010wd  (where more details can be
found). For the scalars, the gauge-invariant fluctuations $\mathfrak{a}^{a}$
obey the following equation

$\displaystyle\Big{[}\mathcal{D}_{r}^{2}+(D-1)\partial_{r}A\,\mathcal{D}_{r}-e^{-2A}q^{2}\Big{]}\mathfrak{a}^{a}-\mathcal{X}^{a}_{\
c}\mathfrak{a}^{c}$ $\displaystyle=$ $\displaystyle 0\,.$ (69)

Here, we used the notation that, given a field $X^{a}$, the sigma-model
covariant, and background-covariant, derivatives are defined by
$D_{b}X^{a}=\partial_{b}X^{a}+\mathcal{G}^{a}{}_{bc}X^{c}$ and
$\mathcal{D}_{r}X^{a}=\partial_{r}X^{a}+\mathcal{G}^{a}{}_{bc}\partial_{r}\Phi^{b}X^{c}$,
respectively, in terms of the sigma-model connection $\mathcal{G}^{a}{}_{bc}$
of Eq. (65). The sigma-model Riemann tensor is then given by
$\mathcal{R}^{a}{}_{bcd}=\partial_{c}\mathcal{G}^{a}{}_{bd}-\partial_{d}\mathcal{G}^{a}{}_{bc}+\mathcal{G}^{a}{}_{ce}\mathcal{G}^{e}{}_{bd}-\mathcal{G}^{a}{}_{de}\mathcal{G}^{e}{}_{bc}$.
The matrix $\mathcal{X}^{a}_{\ c}$ reads as follows:

$\displaystyle\mathcal{X}^{a}_{\ c}$ $\displaystyle\equiv$
$\displaystyle-\mathcal{R}^{a}_{\
bcd}\partial_{r}\Phi^{b}\partial_{r}\Phi^{d}+D_{c}\bigg{(}G^{ab}\frac{\partial\mathcal{V}}{\partial\Phi^{b}}\bigg{)}\,+\frac{4}{(D-2)\partial_{r}A}\bigg{[}\partial_{r}\Phi^{a}\frac{\partial\mathcal{V}}{\partial\Phi^{c}}+G^{ab}\frac{\mathcal{V}}{\partial\Phi^{b}}\partial_{r}\Phi^{d}G_{dc}\bigg{]}$
(70)
$\displaystyle+\frac{16\mathcal{V}}{(D-2)^{2}(\partial_{r}A)^{2}}\partial_{r}\Phi^{a}\partial_{r}\Phi^{b}G_{bc}\,.$

We add infinitely large, boundary-localised mass terms for all the
fluctuations of $\Phi^{a}$, that hence satisfy Dirichlet boundary conditions.
The resulting boundary conditions for the gauge-invariant variables
$\mathfrak{a}^{a}$ are the following:

$\partial_{r}\Phi^{c}\partial_{r}\Phi^{d}G_{db}{\cal
D}_{\rho}\mathfrak{a}^{b}\bigg{|}_{r_{i}}=\left[\frac{(D-2)\partial_{r}A}{2}e^{-2A}q^{2}\delta^{c}_{\
b}+\partial_{r}\Phi^{c}\bigg{(}\frac{4\mathcal{V}}{(D-2)\partial_{r}A}\partial_{r}\Phi^{d}G_{db}+\frac{\partial\mathcal{V}}{\partial\Phi^{b}}\bigg{)}\right]\mathfrak{a}^{b}\bigg{|}_{r_{i}}\,.$
(71)

##  Appendix B Of superpotentials

The associated superpotential, Eq. (6), that is used for example in the
context of holographic renormalisation when $\Delta>5/2$, is not known in
closed form. Yet, we can write it as a power expansion in $\phi$, and we find
that:

$\displaystyle\overline{\mathcal{W}}_{6}$ $\displaystyle=$
$\displaystyle-2-\frac{1}{2}(5-\Delta){\phi}^{2}-\frac{25(2\Delta-5)}{16(4\Delta-15)}{\phi}^{4}-\frac{125(2\Delta-5)\left(4\Delta^{2}-15\Delta+25\right)}{64(4\Delta-15)^{2}(6\Delta-25)}{\phi}^{6}$
$\displaystyle-\frac{625\left(128\Delta^{5}-480\Delta^{4}+800\Delta^{3}-4\times
10^{3}\Delta^{2}+11250\Delta-9375\right)}{1024(4\Delta-15)^{3}\left(48\Delta^{2}-410\Delta+875\right)}\,\phi^{8}$
$\displaystyle-\frac{625}{4096(25-6\Delta)^{2}(15-4\Delta)^{4}\left(16\Delta^{2}-142\Delta+315\right)}\left(3072\Delta^{8}+25600\Delta^{7}-433600\Delta^{6}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+1292\times 10^{3}\Delta^{5}+1065\times
10^{3}\Delta^{4}-10862500\Delta^{3}+18531250\Delta^{2}-12734375\Delta+3515625\right)\,\phi^{10}$
$\displaystyle-\frac{3125}{32768(25-6\Delta)^{2}(4\Delta-15)^{5}\left(192\Delta^{3}-2584\Delta^{2}+11590\Delta-17325\right)}\left(\frac{}{}24576\Delta^{10}+1320960\Delta^{9}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-8153600\Delta^{8}-45968\times
10^{3}\Delta^{7}+473\times 10^{6}\Delta^{6}-12998\times
10^{5}\Delta^{5}+100425\times 10^{4}\Delta^{4}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+133625\times 10^{4}\Delta^{3}-25625\times
10^{5}\Delta^{2}+10^{9}\Delta+87890625\frac{}{}\right)\,\phi^{12}$
$\displaystyle-\frac{15625}{131072(35-8\Delta)^{2}(15-4\Delta)^{6}(6\Delta-25)^{3}\left(336\Delta^{3}-4612\Delta^{2}+21100\Delta-32175\right)}\left(\frac{}{}4718592\Delta^{14}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+714670080\Delta^{13}-2224128\times
10^{3}\Delta^{12}-140363776\times 10^{3}\Delta^{11}+155181568\times
10^{4}\Delta^{10}-54067432\times 10^{5}\Delta^{9}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-6193644\times 10^{6}\Delta^{8}+107537295\times
10^{6}\Delta^{7}-3559741875\times 10^{5}\Delta^{6}+53236421875\times
10^{4}\Delta^{5}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-2431853125\times
10^{5}\Delta^{4}-318709765625\times
10^{3}\Delta^{3}+450370605468750\Delta^{2}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-162176513671875\Delta+823974609375\frac{}{}\right)\,\phi^{14}$
$\displaystyle-\frac{78125}{4194304(35-8\Delta)^{2}(4\Delta-15)^{7}(6\Delta-25)^{3}\left(5376\Delta^{4}-98992\Delta^{3}+683500\Delta^{2}-2097300\Delta+2413125\right)}\times\frac{}{}$
$\displaystyle\frac{}{}\times\left(\frac{}{}150994944\Delta^{16}+57881395200\Delta^{15}+1092472012800\Delta^{14}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-27151204352\times
10^{3}\Delta^{13}+8460660736\times 10^{4}\Delta^{12}+13597392384\times
10^{5}\Delta^{11}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-13484257056\times
10^{6}\Delta^{10}+4795751472\times 10^{7}\Delta^{9}-398284926\times
10^{8}\Delta^{8}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-254111197\times
10^{9}\Delta^{7}+9657946325\times 10^{8}\Delta^{6}-14713569328125\times
10^{5}\Delta^{5}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+88646595703125\times
10^{4}\Delta^{4}+26974482421875\times 10^{4}\Delta^{3}-617424755859375\times
10^{3}\Delta^{2}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+224028259277343750\Delta+61798095703125\frac{}{}\right)\,\phi^{16}$
$\displaystyle-{390625}\left[\frac{}{}16777216(25-6\Delta)^{4}(15-4\Delta)^{8}\left(16\Delta^{2}-142\Delta+315\right)^{2}\times\frac{}{}\right.$
$\displaystyle\left.\frac{}{}\times\left(48384\Delta^{4}-901680\Delta^{3}+6301100\Delta^{2}-19569500\Delta+22790625\right)\right]^{-1}\times\frac{}{}$
$\displaystyle\frac{}{}\times\left(7247757312\Delta^{20}+6263874256896\Delta^{19}+303911927808\times
10^{3}\Delta^{18}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-5902748929228800\Delta^{17}-2758035111936\times
10^{4}\Delta^{16}+13174340632576\times 10^{5}\Delta^{15}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-115787186756608\times
10^{5}\Delta^{14}+31365013166976\times 10^{6}\Delta^{13}+17161524832384\times
10^{7}\Delta^{12}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-18328634352128\times
10^{8}\Delta^{11}+726376592369\times 10^{10}\Delta^{10}-140457265478675\times
10^{8}\Delta^{9}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+3591879533675\times
10^{9}\Delta^{8}+4946038879446875\times
10^{7}\Delta^{7}-126333638659140625\times 10^{6}\Delta^{6}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+1469488973640625\times
10^{8}\Delta^{5}-7281017740966796875\times
10^{4}\Delta^{4}-14333258841552734375\times 10^{3}\Delta^{3}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+322610068359375\times
10^{8}\Delta^{2}-10350940532684326171875\Delta+115871429443359375\right)\,\phi^{18}$
$\displaystyle-{390625}\left[\frac{}{}134217728(25-6\Delta)^{4}(9-2\Delta)^{2}(4\Delta-15)^{9}(8\Delta-35)^{3}\times\frac{}{}\right.$
$\displaystyle\left.\frac{}{}\times\left(193536\Delta^{5}-4526016\Delta^{4}+42336320\Delta^{3}-197998900\Delta^{2}+462983\times
10^{3}\Delta-433021875\right)\frac{}{}\right]^{-1}\times\frac{}{}$
$\displaystyle\frac{}{}\times\left(\frac{}{}463856467968\Delta^{23}+870020787732480\Delta^{22}+93468471475568640\Delta^{21}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-537522325998796800\Delta^{20}-4220626598363136\times
10^{4}\Delta^{19}+70461689518882816\times 10^{4}\Delta^{18}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-24426057346514944\times
10^{5}\Delta^{17}-38568233605083136\times 10^{6}\Delta^{16}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+51863822701075456\times
10^{7}\Delta^{15}-27603786074733376\times 10^{8}\Delta^{14}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+5554033898884608\times
10^{9}\Delta^{13}+1756207389007328\times 10^{10}\Delta^{12}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-1601187067059638\times
10^{11}\Delta^{11}+53057037376123675\times 10^{10}\Delta^{10}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-922419869075040625\times
10^{9}\Delta^{9}+5120596044251203125\times 10^{8}\Delta^{8}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+138753717\times 10^{3}403125\times
10^{10}\Delta^{7}-371419078250947265625\times
10^{7}\Delta^{6}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}+4075653348253369140625\times
10^{6}\Delta^{5}-20041289228924560546875\times
10^{5}\Delta^{4}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-9840867372802734375\times
10^{7}\Delta^{3}+54259196491241455078125\times
10^{4}\Delta^{2}\frac{}{}\right.$
$\displaystyle\left.\frac{}{}-169888076331138610839843750\Delta-60832500457763671875\frac{}{}\right)\,\phi^{20}$
$\displaystyle+\cdots\,.$

We show explicitly terms up to ${\cal O}(\phi^{20})$ in order to allow the
reader to inspect the presence of a pathology in the expansion: for special
choices such as
$\Delta=\frac{15}{4},\frac{25}{6},\frac{35}{8},\frac{9}{2},\frac{55}{12},\frac{65}{14},\frac{75}{16},\frac{85}{18},\frac{19}{4},\cdots$,
some coefficients diverge.

##  Appendix C Asymptotic expansion of the background solutions

In the main body of the paper, we showed the IR expansion for the confining
solutions for any value of $\Delta$, in Eqs. (28), (29), and (30). For the
singular domain-wall solutions, we provided the IR expansions in Eqs. (III.4)
and (III.4). Contrary to the IR expansions, the functional form of the UV
expansions depends explicitly on $\Delta$. In the main body of the paper, we
reported the expansion for $\Delta=3$. In this Appendix, we provide a few
additional representative examples of the UV expansions, expressed as power
series in $z=e^{-{\rho}}$. For simplicity, we set $A_{U}=0=\chi_{U}$.

For $\Delta=1/2$, and hence $\Delta_{J}=1/2$, $\Delta_{V}=9/2$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{1/2}\phi_{J}+z^{9/2}\phi_{V}+\frac{3}{5}z^{11/2}\phi_{J}^{2}\phi_{V}+\frac{35}{192}z^{13/2}\phi_{J}^{4}\phi_{V}+o(z^{7})\,,$
(73) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{z\phi_{J}^{2}}{24}+z^{5}\left(\text{$\chi$5}-\frac{3\phi_{J}\phi_{V}}{100}\right)+z^{6}\left(\frac{25\text{$\chi$5}\phi_{J}^{2}}{48}-\frac{11}{720}\phi_{J}^{3}\phi_{V}\right)$
(74)
$\displaystyle+z^{7}\left(\frac{125\text{$\chi$5}\phi_{J}^{4}}{896}-\frac{65\phi_{J}^{5}\phi_{V}}{16128}\right)+o(z^{7})\,,$
$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{z\phi_{J}^{2}}{6}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{3\phi_{J}\phi_{V}}{25}\right)+z^{6}\left(\frac{25\text{$\chi$5}\phi_{J}^{2}}{192}-\frac{11}{180}\phi_{J}^{3}\phi_{V}\right)$
(75)
$\displaystyle+z^{7}\left(\frac{125\text{$\chi$5}\phi_{J}^{4}}{3584}-\frac{65\phi_{J}^{5}\phi_{V}}{4032}\right)+o(z^{7})\,.$

For $\Delta=1$, and hence $\Delta_{J}=1$, $\Delta_{V}=4$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z\phi_{J}+z^{4}\phi_{V}+\frac{21}{40}z^{6}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(76) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{2}\phi_{J}^{2}+\frac{1}{75}z^{5}\left(75\chi_{5}-4\phi_{J}\phi_{V}\right)-\frac{1}{280}z^{7}\phi_{J}^{2}\left(6\phi_{J}\phi_{V}-125\chi_{5}\right)\,+\,{o}(z^{7})\,,$
(77) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{2}\phi_{J}^{2}+\frac{1}{300}z^{5}\left(75\chi_{5}-64\phi_{J}\phi_{V}\right)+\frac{z^{7}\left(125\chi_{5}\phi_{J}^{2}-96\phi_{J}^{3}\phi_{V}\right)}{1120}\,+\,{o}(z^{7})\,.$
(78)

For $\Delta=3/2$, and hence $\Delta_{J}=3/2$, $\Delta_{V}=7/2$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{3/2}\phi_{J}+z^{7/2}\phi_{V}+\frac{2}{5}z^{13/2}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(79) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{3}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(100\chi_{5}-7\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{7}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(80) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{3}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(25\chi_{5}-28\phi_{J}\phi_{V}\right)-\frac{1}{6}z^{7}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(81)

For $\Delta=17/10$, and hence $\Delta_{J}=17/10$, $\Delta_{V}=33/10$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{17/10}\phi_{J}+z^{33/10}\phi_{V}+\frac{42}{125}z^{67/10}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(82) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{17/5}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{187\phi_{J}\phi_{V}}{2500}\right)-\frac{1}{24}z^{33/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(83) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{17/5}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{187\phi_{J}\phi_{V}}{625}\right)-\frac{1}{6}z^{33/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(84)

For $\Delta=7/4$, and hence $\Delta_{J}=7/4$, $\Delta_{V}=13/4$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{7/4}\phi_{J}+z^{13/4}\phi_{V}+\frac{51}{160}z^{27/4}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(85) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{7/2}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{91\phi_{J}\phi_{V}}{1200}\right){-\frac{1}{24}z^{13/2}\phi_{V}^{2}}\,+\,{o}(z^{7})\,,$
(86) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{7/2}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{91\phi_{J}\phi_{V}}{300}\right){-\frac{1}{6}z^{13/2}\phi_{V}^{2}}\,+\,{o}(z^{7})\,.$
(87)

For $\Delta=9/5$, and hence $\Delta_{J}=9/5$, $\Delta_{V}=16/5$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{9/5}\phi_{J}+z^{16/5}\phi_{V}+\frac{301z^{34/5}\phi_{J}^{2}\phi_{V}}{1000}\,+\,{o}(z^{7})\,,$
(88) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{18/5}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{48\phi_{J}\phi_{V}}{625}\right)-\frac{1}{24}z^{32/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(89) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{18/5}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{192\phi_{J}\phi_{V}}{625}\right)-\frac{1}{6}z^{32/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(90)

For $\Delta=37/20$, and hence $\Delta_{J}=37/20$, $\Delta_{V}=63/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{37/20}\phi_{J}+z^{63/20}\phi_{V}+\frac{1131z^{137/20}\phi_{J}^{2}\phi_{V}}{4000}\,+\,{o}(z^{7})\,,$
(91) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{37/10}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{777\phi_{J}\phi_{V}}{10000}\right)-\frac{1}{24}z^{63/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(92) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{37/10}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{777\phi_{J}\phi_{V}}{2500}\right)-\frac{1}{6}z^{63/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(93)

For $\Delta=19/10$, and hence $\Delta_{J}=19/10$, $\Delta_{V}=31/10$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{19/10}\phi_{J}+{z^{31/10}\phi_{V}+\frac{33}{125}z^{69/10}\phi_{J}^{2}\phi_{V}}\,+\,{o}(z^{7})\,,$
(94) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{19/5}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{589\phi_{J}\phi_{V}}{7500}\right)-\frac{1}{24}z^{31/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(95) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{19/5}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{589\phi_{J}\phi_{V}}{1875}\right)-\frac{1}{6}z^{31/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(96)

For $\Delta=39/20$, and hence $\Delta_{J}=39/20$, $\Delta_{V}=61/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{39/20}\phi_{J}+z^{61/20}\phi_{V}+\frac{979z^{139/20}\phi_{J}^{2}\phi_{V}}{4000}+\,{o}(z^{7})\,,$
(97) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{39/10}\phi_{J}^{2}+z^{5}\left(\chi_{5}-\frac{793\phi_{J}\phi_{V}}{10000}\right)-\frac{1}{24}z^{61/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(98) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{39/10}\phi_{J}^{2}+z^{5}\left(\frac{\chi_{5}}{4}-\frac{793\phi_{J}\phi_{V}}{2500}\right)-\frac{1}{6}z^{61/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(99)

For $\Delta=2$, and hence $\Delta_{J}=2$, $\Delta_{V}=3$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{2}\phi_{J}+z^{3}\phi_{\Delta_{V}}+\frac{9}{40}z^{7}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(100) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{4}\phi_{J}^{2}+\frac{1}{25}z^{5}\left(25\chi_{5}-2\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{6}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(101) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{4}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(25\chi_{5}-32\phi_{J}\phi_{V}\right)-\frac{1}{6}z^{6}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(102)

For $\Delta=9/4$, and hence $\Delta_{J}=9/4$, $\Delta_{V}=11/4$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{9/4}\phi_{J}+z^{11/4}\phi_{V}\,+\,{o}(z^{7})\,,$ (103) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{9/2}\phi_{J}^{2}+\frac{1}{400}z^{5}\left(400\chi_{5}-33\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{11/2}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(104) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{9/2}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(25\chi_{5}-33\phi_{J}\phi_{V}\right)-\frac{1}{6}z^{11/2}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(105)

For $\Delta=23/10$, and hence $\Delta_{J}=23/10$, $\Delta_{V}=27/10$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{23/10}\phi_{J}+z^{27/10}\phi_{V}\,+\,{o}(z^{7})\,,$ (106)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{23/5}\phi_{J}^{2}+\frac{z^{5}\left(2500\chi_{5}-207\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{24}z^{27/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(107) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{23/5}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-828\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{27/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(108)

For $\Delta=47/20$, and hence $\Delta_{J}=47/20$, $\Delta_{V}=53/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{47/20}\phi_{J}+z^{53/20}\phi_{V}\,+\,{o}(z^{7})\,,$ (109)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{47/10}\phi_{J}^{2}+\frac{z^{5}\left(30000\chi_{5}-2491\phi_{J}\phi_{V}\right)}{30000}-\frac{1}{24}z^{53/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(110) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{47/10}\phi_{J}^{2}+\frac{z^{5}\left(1875\chi_{5}-2491\phi_{J}\phi_{V}\right)}{7500}-\frac{1}{6}z^{53/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(111)

For $\Delta=12/5$, and hence $\Delta_{J}=12/5$, $\Delta_{V}=13/5$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{12/5}\phi_{J}+z^{13/5}\phi_{V}\,+\,{o}(z^{7})\,,$ (112) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{24/5}\phi_{J}^{2}+\frac{1}{625}z^{5}\left(625\chi_{5}-52\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{26/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(113) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{24/5}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-832\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{26/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(114)

For $\Delta=49/20$, and hence $\Delta_{J}=49/20$, $\Delta_{V}=51/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{49/20}\phi_{J}+z^{51/20}\phi_{V}\,+\,{o}(z^{7})\,,$ (115)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{49/10}\phi_{J}^{2}+\frac{z^{5}\left(10000\chi_{5}-833\phi_{J}\phi_{V}\right)}{10000}-\frac{1}{24}z^{51/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(116) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{49/10}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-833\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{51/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(117)

For $\Delta=249/100$, and hence $\Delta_{J}=249/100$, $\Delta_{V}=251/100$, we
have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{249/100}\phi_{J}+z^{251/100}\phi_{V}+o(z^{7})\,,$ (118) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{249/50}\phi_{J}^{2}+z^{5}\left(\text{$\chi$5}-\frac{20833\phi_{J}\phi_{V}}{250000}\right)-\frac{1}{24}z^{251/50}\phi_{V}^{2}+o(z^{7})\,,$
(119) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{249/50}\phi_{J}^{2}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{20833\phi_{J}\phi_{V}}{62500}\right)-\frac{1}{6}z^{251/50}\phi_{V}^{2}+o(z^{7})\,.$
(120)

For $\Delta=2499/1000$, and hence $\Delta_{J}=2499/1000$,
$\Delta_{V}=2501/1000$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{2499/1000}\phi_{J}+z^{2501/1000}\phi_{V}+o(z^{7})\,,$ (121)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{2499/500}\phi_{J}^{2}+z^{5}\left(\text{$\chi$5}-\frac{2083333\phi_{J}\phi_{V}}{25000000}\right)-\frac{1}{24}z^{2501/500}\phi_{V}^{2}+o(z^{7})\,,$
(122) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{2499/500}\phi_{J}^{2}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{2083333\phi_{J}\phi_{V}}{6250000}\right)-\frac{1}{6}z^{2501/500}\phi_{V}^{2}+o(z^{7})\,.$
(123)

For $\Delta=5/2$, and hence $\Delta_{J}=\Delta_{V}=5/2$, we have

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi_{V}z^{5/2}+\phi_{J}z^{5/2}\log(z)+\frac{1}{200}\left(50\phi_{V}^{2}\phi_{J}-25\phi_{V}\phi_{J}^{2}+8\phi_{J}^{3}\right)z^{15/2}+$
(124)
$\displaystyle+\frac{1}{8}\left(4\phi_{V}\phi_{J}^{2}-\phi_{J}^{3}\right)z^{15/2}\log(z)+\frac{1}{4}\phi_{J}^{3}z^{15/2}\log^{2}(z)\,+\,{o}(z^{8})\,,$
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}+\left(\chi_{5}+\frac{1}{600}(-25\phi_{V}^{2}-2\phi_{J}^{2})\right)z^{5}+$
(125)
$\displaystyle-\frac{1}{12}\phi_{V}\phi_{J}z^{5}\log(z)-\frac{1}{24}\phi_{J}^{2}z^{5}\log^{2}(z)\,+\,{o}(z^{8})\,,$
$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}+\frac{1}{300}\left(75\chi_{5}-50\phi_{V}^{2}-4\phi_{J}^{2}\right)z^{5}-\frac{1}{3}\phi_{V}\phi_{J}z^{5}\log(z)-\frac{1}{6}\phi_{J}^{2}z^{5}\log^{2}(z)\,+\,{o}(z^{8})\,.$
(126)

For $\Delta=2501/1000$, and hence $\Delta_{J}=2499/1000$,
$\Delta_{V}=2501/1000$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{2499/1000}\phi_{J}+z^{2501/1000}\phi_{V}+o(z^{7})\,,$ (127)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{2499/500}\phi_{J}^{2}+z^{5}\left(\text{$\chi$5}-\frac{2083333\phi_{J}\phi_{V}}{25000000}\right)-\frac{1}{24}z^{2501/500}\phi_{V}^{2}+o(z^{7})\,,$
(128) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{2499/500}\phi_{J}^{2}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{2083333\phi_{J}\phi_{V}}{6250000}\right)-\frac{1}{6}z^{2501/500}\phi_{V}^{2}+o(z^{7})\,.$
(129)

For $\Delta=251/100$, and hence $\Delta_{J}=249/100$, $\Delta_{V}=251/100$, we
have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{249/100}\phi_{J}+z^{251/100}\phi_{V}+o(z^{7})\,,$ (130) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{249/50}\phi_{J}^{2}+z^{5}\left(\text{$\chi$5}-\frac{20833\phi_{J}\phi_{V}}{250000}\right)-\frac{1}{24}z^{251/50}\phi_{V}^{2}+o(z^{7})\,,$
(131) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{249/50}\phi_{J}^{2}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{20833\phi_{J}\phi_{V}}{62500}\right)-\frac{1}{6}z^{251/50}\phi_{V}^{2}+o(z^{7})\,.$
(132)

For $\Delta=51/20$, and hence $\Delta_{J}=49/20$, $\Delta_{V}=51/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{49/20}\phi_{J}+z^{51/20}\phi_{V}\,+\,{o}(z^{7})\,,$ (133)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{49/10}\phi_{J}^{2}+\frac{z^{5}\left(10000\chi_{5}-833\phi_{J}\phi_{V}\right)}{10000}-\frac{1}{24}z^{51/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(134) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{49/10}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-833\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{51/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(135)

For $\Delta=13/5$, and hence $\Delta_{J}=12/5$, $\Delta_{V}=13/5$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{12/5}\phi_{J}+z^{13/5}\phi_{V}\,+\,{o}(z^{7})\,,$ (136) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{24/5}\phi_{J}^{2}+\frac{1}{625}z^{5}\left(625\chi_{5}-52\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{26/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(137) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{24/5}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-832\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{26/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(138)

For $\Delta=53/20$, and hence $\Delta_{J}=47/20$, $\Delta_{V}=53/20$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{47/20}\phi_{J}+z^{53/20}\phi_{V}\,+\,{o}(z^{7})\,,$ (139)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{47/10}\phi_{J}^{2}+\frac{z^{5}\left(30000\chi_{5}-2491\phi_{J}\phi_{V}\right)}{30000}-\frac{1}{24}z^{53/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(140) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{47/10}\phi_{J}^{2}+\frac{z^{5}\left(1875\chi_{5}-2491\phi_{J}\phi_{V}\right)}{7500}-\frac{1}{6}z^{53/10}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(141)

For $\Delta=27/10$, and hence $\Delta_{J}=23/10$, $\Delta_{V}=27/10$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{23/10}\phi_{J}+z^{27/10}\phi_{V}-\frac{125}{966}z^{69/10}\phi_{J}^{3}\,+\,{o}(z^{7})\,,$
(142) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{23/5}\phi_{J}^{2}+\frac{z^{5}\left(2500\chi_{5}-207\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{24}z^{27/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(143) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{23/5}\phi_{J}^{2}+\frac{z^{5}\left(625\chi_{5}-828\phi_{J}\phi_{V}\right)}{2500}-\frac{1}{6}z^{27/5}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(144)

For $\Delta=11/4$, and hence $\Delta_{J}=9/4$, $\Delta_{V}=11/4$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{9/4}\phi_{J}+z^{11/4}\phi_{V}-\frac{25}{144}z^{27/4}\phi_{J}^{3}\,+\,{o}(z^{7})\,,$
(145) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{9/2}\phi_{J}^{2}+\frac{1}{400}z^{5}\left(400\chi_{5}-33\phi_{J}\phi_{V}\right)-\frac{1}{24}z^{11/2}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(146) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{9/2}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(25\chi_{5}-33\phi_{J}\phi_{V}\right)-\frac{1}{6}z^{11/2}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(147)

For $\Delta=3$, and hence $\Delta_{J}=2$, $\Delta_{V}=3$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{2}\phi_{J}+z^{3}\phi_{V}-\frac{25}{48}z^{6}\phi_{J}^{3}-\frac{57}{80}z^{7}\left(\phi_{J}^{2}\phi_{V}\right)+o(z^{7})\,,$
(148) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{4}\phi_{J}^{2}+z^{5}\left(\text{$\chi$5}-\frac{2\phi_{J}\phi_{V}}{25}\right)-\frac{1}{24}z^{6}\phi_{V}^{2}+o(z^{7})\,,$
(149) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{4}\phi_{J}^{2}+z^{5}\left(\frac{\text{$\chi$5}}{4}-\frac{8\phi_{J}\phi_{V}}{25}\right)-\frac{1}{6}z^{6}\phi_{V}^{2}+o(z^{7})\,.$
(150)

For $\Delta=7/2$, and hence $\Delta_{J}=3/2$, $\Delta_{V}=7/2$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{3/2}\phi_{J}+z^{7/2}\phi_{V}-\frac{25}{6}z^{9/2}\phi_{J}^{3}-\frac{21}{10}z^{13/2}\phi_{J}^{2}\phi_{V}\,+\,{o}(z^{7})\,,$
(151) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{3}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(100\chi_{5}-7\phi_{J}\phi_{V}\right)+\frac{25}{96}z^{6}\phi_{J}^{4}-\frac{1}{24}z^{7}\phi_{V}^{2}\,+\,{o}(z^{7})\,,$
(152) $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}{-\frac{1}{6}z^{3}\phi_{J}^{2}+\frac{1}{100}z^{5}\left(25\chi_{5}-28\phi_{J}\phi_{V}\right)}+\frac{25}{24}z^{6}\phi_{J}^{4}-\frac{1}{6}z^{7}\phi_{V}^{2}\,+\,{o}(z^{7})\,.$
(153)

For $\Delta=4$, and hence $\Delta_{J}=1$, $\Delta_{V}=4$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z\phi_{J}+\frac{75}{8}z^{3}\phi_{J}^{3}+z^{4}\phi_{V}-\frac{7875}{64}z^{5}\phi_{J}^{5}-\frac{51}{10}z^{6}\phi_{J}^{2}\phi_{V}+\frac{183125z^{7}\phi_{J}^{7}}{1536}\,+\,{o}(z^{7})\,,$
(154) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}z^{2}\phi_{J}^{2}-\frac{75}{128}z^{4}\phi_{J}^{4}+\frac{1}{75}z^{5}\left(75\chi_{5}-4\phi_{J}\phi_{V}\right)+\frac{3125z^{6}\phi_{J}^{6}}{1536}$
$\displaystyle+\frac{1}{280}z^{7}\left(125\chi_{5}\phi_{J}^{2}-156\phi_{J}^{3}\phi_{V}\right)\,+\,{o}(z^{7})\,,$
$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}z^{2}\phi_{J}^{2}-\frac{75}{32}z^{4}\phi_{J}^{4}+\frac{1}{300}z^{5}\left(75\chi_{5}-64\phi_{J}\phi_{V}\right)+\frac{3125}{384}z^{6}\phi_{J}^{6}$
$\displaystyle+\frac{z^{7}\left(125\chi_{5}\phi_{J}^{2}-2496\phi_{J}^{3}\phi_{V}\right)}{1120}\,+\,{o}(z^{7})\,.$

For $\Delta=24/5$, and hence $\Delta_{J}=1/5$, $\Delta_{V}=24/5$, we have

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
z^{1/5}\phi_{J}+\frac{2875}{168}z^{3/5}\phi_{J}^{3}+\frac{589375z\phi_{J}^{5}}{1216}+\frac{1194097540625z^{7/5}\phi_{J}^{7}}{72930816}+\frac{40083780796875z^{9/5}\phi_{J}^{9}}{64827392}$
(157)
$\displaystyle+\frac{206305363012094528125z^{11/5}\phi_{J}^{11}}{8070232375296}+\frac{51555188135555851650546875z^{13/5}\phi_{J}^{13}}{44741368288641024}$
$\displaystyle+\frac{220895018700707881287319140625z^{3}\phi_{J}^{15}}{3872162055526023168}+\frac{143635271586199611133308619908203125z^{17/5}\phi_{J}^{17}}{45319784697876575158272}$
$\displaystyle+\frac{2030227399573934911294696548892509765625z^{19/5}\phi_{J}^{19}}{9789073494741340234186752}$
$\displaystyle+\frac{62734499516892742142454970327066041783203125z^{21/5}\phi_{J}^{21}}{3471858066134928669724901376}$
$\displaystyle+\frac{703737750200907787167436115783648540622242333984375z^{23/5}\phi_{J}^{23}}{202562087010576278306429645881344})$
$\displaystyle+z^{24/5}\phi_{V}-\frac{42889968257354026233434739048798684916940287126220703125z^{5}\phi_{J}^{25}}{102091291853330444266440541524197376}$
$\displaystyle-\frac{5313}{125}z^{26/5}\left(\phi_{J}^{2}\phi_{V}\right)+\frac{40372906759830069883219591463928127456136451751634521484375z^{27/5}\phi_{J}^{27}}{5811221188951714835775331565278593024}$
$\displaystyle+\frac{512647z^{28/5}\phi_{J}^{4}\phi_{V}}{3024}+\frac{23451902891374040266318476466845032509323071584144748101806640625z^{29/5}\phi_{J}^{29}}{1344344664967333903248922102985288595800064}$
$\displaystyle+\frac{2812375255z^{6}\phi_{J}^{6}\phi_{V}}{6664896}-\frac{9356448978981275z^{32/5}\left(\phi_{J}^{8}\phi_{V}\right)}{786777643008}$
$\displaystyle+\frac{3382994549224018067073083754105413390854841531398992869892655029296875z^{31/5}\phi_{J}^{31}}{12873444511727189457511678058187123593381412864}$
$\displaystyle+\frac{63252672595173277436174439987322783944040397901181003729679595947265625z^{33/5}\phi_{J}^{33}}{11443061788201946184454824940610776527450144768}$
$\displaystyle-\frac{882653673174311913125z^{34/5}\left(\phi_{J}^{10}\phi_{V}\right)}{1090473813209088}\,+\,{\mathcal{O}(z^{7})}\,,$
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-\frac{\log(z)}{3}-\frac{1}{24}\phi_{J}^{2}z^{2/5}-\frac{2875\phi_{J}^{4}z^{4/5}}{2688}-\frac{445840625\phi_{J}^{6}z^{6/5}}{12870144}-\frac{1245177090625\phi_{J}^{8}z^{8/5}}{1000194048}$
(158)
$\displaystyle-\frac{12757500935258125\phi_{J}^{10}z^{2}}{266051616768}-\frac{252462758077427834375\phi_{J}^{12}z^{12/5}}{129123718004736}$
$\displaystyle-\frac{219400845248171582865859375\phi_{J}^{14}z^{14/5}}{2607782608823648256}-\frac{595634988671267098623623224609375\phi_{J}^{16}z^{16/5}}{155382118964148257685504}$
$\displaystyle-\frac{37370923454539023891704902220703125\phi_{J}^{18}z^{18/5}}{199777010096762045595648}-\frac{3005589851661754675633072282749116796875\phi_{J}^{20}z^{4}}{299011699475735483516977152}$
$\displaystyle-\frac{390603999209272169612305206890628037373755859375\phi_{J}^{22}z^{22/5}}{607686261031728834919288937644032}$
$\displaystyle-\frac{1396398597179619157280207994845812226729262841796875\phi_{J}^{24}z^{24/5}}{19445960353015322717417246004609024}$
$\displaystyle+\left(-\frac{1}{625}(8\phi_{J}\phi_{V})+\chi_{5}\right)z^{5}+\frac{1306083301937461832252797063598660554434038579290771484375\phi_{J}^{26}z^{26/5}}{605197178106542873611459530155442044928}$
$\displaystyle+\left(-\frac{4117\phi_{J}^{3}\phi_{V}}{70875}+\frac{125\phi_{J}^{2}\chi_{5}}{216}\right)z^{27/5}+\frac{5\phi_{J}^{4}(-9784936\phi_{J}\phi_{V}+55981125\chi_{5})z^{29/5}}{19994688}$
$\displaystyle+\frac{13030865310005017933330153469549477844134773156086490478515625\phi_{J}^{28}z^{28/5}}{474474587635529612911384271641866563223552}$
$\displaystyle+\frac{7032769665707387278806255307823137953470132552224159218505859375\phi_{J}^{30}z^{6}}{10754757319738671225991376823882308766400512}$
$\displaystyle-\frac{25\left(\phi_{J}^{6}(2692663154104\phi_{J}\phi_{V}-10082616060375\chi_{5})\right)z^{31/5}}{590083232256}$
$\displaystyle+\frac{5540969477438023313564601832074044301054003663389867568262069091796875\phi_{J}^{32}z^{32/5}}{290788393676661220687322610255520909403438972928}$
$\displaystyle-\frac{25\left(\phi_{J}^{8}(35613091214958616\phi_{J}\phi_{V}-90309329147491875\chi_{5})\right)z^{33/5}}{155781973315584}$
$\displaystyle+\frac{181830278970246170965622209207842711341265597685817924668243991119384765625\phi_{J}^{34}z^{34/5}}{294132460203942824725226820273459399861578521116672}$
$\displaystyle\,+\,{\mathcal{O}(z^{7})}\,,$ $\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{4\log(z)}{3}-\frac{1}{6}\phi_{J}^{2}z^{2/5}-\frac{2875}{672}\phi_{J}^{4}z^{4/5}-\frac{445840625\phi_{J}^{6}z^{6/5}}{3217536}-\frac{1245177090625\phi_{J}^{8}z^{8/5}}{250048512}$
(159)
$\displaystyle-\frac{12757500935258125\phi_{J}^{10}z^{2}}{66512904192}-\frac{252462758077427834375\phi_{J}^{12}z^{12/5}}{32280929501184}$
$\displaystyle-\frac{219400845248171582865859375\phi_{J}^{14}z^{14/5}}{651945652205912064}-\frac{595634988671267098623623224609375\phi_{J}^{16}z^{16/5}}{38845529741037064421376}$
$\displaystyle-\frac{37370923454539023891704902220703125\phi_{J}^{18}z^{18/5}}{49944252524190511398912}-\frac{3005589851661754675633072282749116796875\phi_{J}^{20}z^{4}}{74752924868933870879244288}$
$\displaystyle-\frac{390603999209272169612305206890628037373755859375\phi_{J}^{22}z^{22/5}}{151921565257932208729822234411008}$
$\displaystyle-\frac{1396398597179619157280207994845812226729262841796875\phi_{J}^{24}z^{24/5}}{4861490088253830679354311501152256}$
$\displaystyle+\left(-\frac{1}{625}(32\phi_{J}\phi_{V})+\frac{\chi_{5}}{4}\right)z^{5}+\frac{1306083301937461832252797063598660554434038579290771484375\phi_{J}^{26}z^{26/5}}{151299294526635718402864882538860511232}$
$\displaystyle+\left(-\frac{16468\phi_{J}^{3}\phi_{V}}{70875}+\frac{125\phi_{J}^{2}\chi_{5}}{864}\right)z^{27/5}+\left(-\frac{873655\phi_{J}^{5}\phi_{V}}{89262}+\frac{181875\phi_{J}^{4}\chi_{5}}{51968}\right)z^{29/5}$
$\displaystyle+\frac{13030865310005017933330153469549477844134773156086490478515625\phi_{J}^{28}z^{28/5}}{118618646908882403227846067910466640805888}$
$\displaystyle+\frac{7032769665707387278806255307823137953470132552224159218505859375\phi_{J}^{30}z^{6}}{2688689329934667806497844205970577191600128}$
$\displaystyle+\frac{25\phi_{J}^{6}(-43082610465664\phi_{J}\phi_{V}+10082616060375\chi_{5})z^{31/5}}{2360332929024}$
$\displaystyle+\frac{5540969477438023313564601832074044301054003663389867568262069091796875\phi_{J}^{32}z^{32/5}}{72697098419165305171830652563880227350859743232}$
$\displaystyle-\frac{25\left(\phi_{J}^{8}(569809459439337856\phi_{J}\phi_{V}-90309329147491875\chi_{5})\right)z^{33/5}}{623127893262336}$
$\displaystyle+\frac{181830278970246170965622209207842711341265597685817924668243991119384765625\phi_{J}^{34}z^{34/5}}{73533115050985706181306705068364849965394630279168}$
$\displaystyle\,+\,{\mathcal{O}(z^{7})}\,.$

##  Appendix D Asymptotic expansions of the fluctuations

As described in the body of the paper, in order to improve the numerical
convergence of the spectra of fluctuations, we write the asymptotic expansions
of such solutions, both in the IR and in the UV. For the UV case, the
expansion depends crucially on $\Delta$; while we did compute it explicitly
for all the values of $\Delta$ used in the numerical study, we report here
only one example, for illustrative purposes.

###  D.1 IR expansions

For convenience, we put $\rho_{o}=0$ and $A_{I}=0$ in this subsection,222The
dependence on $\rho_{o}$ and $A_{I}$ can be reinstated by making the
substitutions $\rho\rightarrow\rho-\rho_{o}$ and $q^{2}\rightarrow
e^{-2A_{I}}q^{2}$ in the expressions. while $\chi_{I}=0$ in order to avoid a
conical singularity, as explained in the main text. For the fluctuations of
the scalars $\Phi^{a}=\\{{\phi},\chi\\}$, we have

$\displaystyle\mathfrak{a}^{1}=$ $\displaystyle\
\mathfrak{a}^{1}_{I,0}+\mathfrak{a}^{1}_{I,l}\log(\rho)+\frac{1}{4}\rho^{2}\bigg{[}-\frac{1}{4}\Delta\left(\mathfrak{a}^{1}_{I,0}\left(\Delta\left(15\phi_{I}^{2}-4\right)+20\right)+6\phi_{I}(\mathfrak{a}^{2}_{I,0}-\mathfrak{a}^{2}_{I,l})\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)\right)$
$\displaystyle+q^{2}(\mathfrak{a}^{1}_{I,0}-\mathfrak{a}^{1}_{I,l})-\frac{1}{48}\mathfrak{a}^{1}_{I,l}\left(\Delta\left(25\Delta\phi_{I}^{4}+20(10-11\Delta)\phi_{I}^{2}+48(\Delta-5)\right)+400\right)$
$\displaystyle+\log(\rho)\left(\mathfrak{a}^{1}_{I,l}\left(-\frac{15\Delta^{2}\phi_{I}^{2}}{4}+(\Delta-5)\Delta+q^{2}\right)-\frac{3}{2}\mathfrak{a}^{2}_{I,l}\Delta\phi_{I}\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)\right)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(160) $\displaystyle\mathfrak{a}^{2}=$ $\displaystyle\
\mathfrak{a}^{2}_{I,0}+\mathfrak{a}^{2}_{I,l}\log(\rho)+\frac{1}{4}\rho^{2}\bigg{[}-\frac{1}{4}\Delta\phi_{I}(\mathfrak{a}^{1}_{I,0}-\mathfrak{a}^{1}_{I,l})\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)+q^{2}(\mathfrak{a}^{2}_{I,0}-\mathfrak{a}^{2}_{I,l})$
$\displaystyle-\frac{3}{8}\mathfrak{a}^{2}_{I,0}\left(\Delta\phi_{I}^{2}\left(\Delta\left(5\phi_{I}^{2}-8\right)+40\right)+80\right)+\frac{13}{48}\mathfrak{a}^{2}_{I,l}\left(\Delta\phi_{I}^{2}\left(\Delta\left(5\phi_{I}^{2}-8\right)+40\right)+80\right)$
$\displaystyle+\log(\rho)\left(-\frac{5}{4}\mathfrak{a}^{1}_{I,l}\Delta^{2}\phi_{I}^{3}+\mathfrak{a}^{1}_{I,l}(\Delta-5)\Delta\phi_{I}+\mathfrak{a}^{2}_{I,l}\left(-\frac{15}{8}\Delta^{2}\phi_{I}^{4}+3(\Delta-5)\Delta\phi_{I}^{2}+q^{2}-30\right)\right)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(161)

For the fluctuations of the vector $\chi_{M}$, we have

$\displaystyle\mathfrak{v}=$ $\displaystyle\
\mathfrak{v}_{I,-2}\rho^{-2}+\frac{1}{2}q^{2}\mathfrak{v}_{I,-2}\log(\rho)+\mathfrak{v}_{I,0}+\frac{1}{12288}\rho^{2}\Big{[}1536q^{2}\mathfrak{v}_{I,0}+80\Delta^{2}\mathfrak{v}_{I,-2}\phi_{I}^{4}\left(2\left(8\Delta^{2}-50\Delta+75\right)-3q^{2}\right)$
$\displaystyle+128(\Delta-5)\Delta\mathfrak{v}_{I,-2}\phi_{I}^{2}\left(-3(\Delta-5)\Delta+3q^{2}-50\right)-64\left(9q^{4}+60q^{2}-500\right)\mathfrak{v}_{I,-2}+125\Delta^{4}\mathfrak{v}_{I,-2}\phi_{I}^{8}$
$\displaystyle-1000(\Delta-2)\Delta^{3}\mathfrak{v}_{I,-2}\phi_{I}^{6}+768q^{4}\mathfrak{v}_{I,-2}\log(\rho)\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(162)

For the tensor fluctuations $\mathfrak{e}^{\mu}_{\ \nu}$, we have

$\mathfrak{e}=\mathfrak{e}_{I,0}+\mathfrak{e}_{I,l}\log(\rho)+\frac{1}{192}\rho^{2}\Big{[}48q^{2}(\mathfrak{e}_{I,0}-\mathfrak{e}_{I,l})-25\Delta^{2}\mathfrak{e}_{I,l}\phi_{I}^{4}+40(\Delta-5)\Delta\mathfrak{e}_{I,l}\phi_{I}^{2}-400\mathfrak{e}_{I,l}+48\mathfrak{e}_{I,l}q^{2}\log(\rho)\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(163)

###  D.2 UV expansions

In this subsection, we put $\Delta=3$, and $A_{U}=0=\chi_{U}$.333The
dependence on $\chi_{U}$ and $A_{U}$ can be reinstated by making the
substitution $q^{2}\rightarrow e^{2\chi_{U}-2A_{U}}q^{2}$ in the expressions.
We write the expansions in terms of $z\equiv e^{-\rho}$.

For the fluctuations of the scalars, we have

$\displaystyle\mathfrak{a}^{1}=$ $\displaystyle\
\mathfrak{a}^{1}_{2}z^{2}+\mathfrak{a}^{1}_{3}z^{3}+\frac{1}{2}\mathfrak{a}^{1}_{2}q^{2}z^{4}+\frac{1}{6}\mathfrak{a}^{1}_{3}q^{2}z^{5}+\frac{1}{48}\mathfrak{a}^{1}_{2}\left(2q^{4}-99\phi_{J}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,,$
(164) $\displaystyle\mathfrak{a}^{2}=$ $\displaystyle\
\mathfrak{a}^{2}_{0}-\frac{1}{6}\mathfrak{a}^{2}_{0}q^{2}z^{2}+\frac{1}{24}\mathfrak{a}^{2}_{0}q^{4}z^{4}+\mathfrak{a}^{2}_{5}z^{5}+\frac{1}{144}\mathfrak{a}^{2}_{0}q^{2}\left(q^{4}-14\phi_{J}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,.$
(165)

For the vector fluctuations, we have

$\displaystyle\mathfrak{v}=$ $\displaystyle\
\mathfrak{v}_{0}-\frac{1}{6}q^{2}\mathfrak{v}_{0}z^{2}+\frac{1}{24}q^{4}\mathfrak{v}_{0}z^{4}+\mathfrak{v}_{5}z^{5}+\frac{1}{144}q^{2}\mathfrak{v}_{0}\left(q^{4}-14\phi_{J}^{2}\right)z^{6}$
$\displaystyle+\frac{1}{70}q^{2}(70\mathfrak{v}_{0}\chi_{5}-2\mathfrak{v}_{0}\phi_{J}\phi_{V}+5\mathfrak{v}_{5})z^{7}+\mathcal{O}\left(z^{8}\right)\,.$
(166)

For the tensor fluctuations, we have

$\mathfrak{e}=\mathfrak{e}_{0}-\frac{1}{6}\mathfrak{e}_{0}q^{2}z^{2}+\frac{1}{24}\mathfrak{e}_{0}q^{4}z^{4}+\mathfrak{e}_{5}z^{5}+\mathcal{O}\left(z^{6}\right)\,.$
(167)

##  Appendix E Probe approximation

In this appendix, we further discuss the probe approximation. More details can
be found in Ref. Elander:2020csd . The fluctuations
$\mathfrak{a}^{a}=\varphi^{a}\,-\,\frac{\partial_{r}\Phi^{a}}{6\partial_{r}A}h$
are formed by the mixing of the fluctuations of the scalars
$\Phi^{a}=({\phi},\chi)$ and fluctuations $h$ of the trace of the four-
dimensional part of the metric. The latter couples to the trace of the stress-
energy tensor of the boundary theory. If $\mathfrak{a}^{a}$ is mostly composed
of $h$, hence
$\mathfrak{a}^{a}\simeq\frac{\partial_{r}\Phi^{a}}{6\partial_{r}A}h$, the
couplings of the states sourced by the operators at the boundary are well
approximated by the dilatonic counterparts. Conversely, in the probe
approximation for computing spectra, one neglects the mixing between the
scalar and metric fluctuations. This will lead to correct results only if
$\mathfrak{a}^{a}\simeq\varphi^{a}$, such that the contribution of $h$ in Eq.
(69) can be neglected.

The probe approximation is hence obtained by neglecting the contributions from
the metric fluctuations, in particular $h$, in Eqs. (69) and (71), leading to
Elander:2020csd

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left[\frac{}{}{\cal
D}_{r}^{2}+(D-1)\partial_{r}A\,{\cal
D}_{r}-e^{-2A}q^{2}\right]\mathfrak{p}^{a}-\left[\frac{}{}V^{a}_{\,\,\,\,|c}-{\cal
R}^{a}_{\,\,\,\,bcd}\partial_{r}\Phi^{b}\partial_{r}\Phi^{d}\right]\mathfrak{p}^{c}\,,$
(168)

for the equations of motion, while the boundary conditions reduce to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left.\frac{}{}\mathfrak{p}^{a}\right|_{r_{i}}\,.$ (169)

In these equations, we have replaced $\mathfrak{a}^{a}$ with the probe
fluctuations that we denote by $\mathfrak{p}^{a}$.

We perform the calculation of the spectra of scalar fluctuations in two ways
for $\Delta=5/2$ in the body of the paper. First, by solving the exact Eqs.
(69), with the boundary conditions in Eqs. (71), and finding the spectrum of
masses. Then, we repeat the calculation for the same background, but using the
probe approximation and solving Eqs. (168), with the boundary conditions in
Eqs. (169). If the two processes result in different spectra, overlap with the
dilaton can not be neglected. In the next subsections, IR and UV expansions
for fluctuations in the probe approximation are given.

###  E.1 IR expansions

In this subsection, we put $\Delta=5/2$. As in Appendix D, we put $\rho_{o}=0$
and $A_{I}=0$, while $\chi_{I}=0$ in order to avoid a conical singularity. For
the fluctuations of the scalars in the probe approximation, we have

$\displaystyle\mathfrak{p}^{1}=\,$
$\displaystyle\mathfrak{p}^{1}_{I,0}+\mathfrak{p}^{1}_{I,l}\log(\rho)+\rho^{2}\bigg{[}\Delta\left(-\frac{5\mathfrak{p}^{2}_{I,l}\phi_{I}}{2}+\frac{5\mathfrak{p}^{2}_{I,0}\phi_{I}}{2}\right)+\Delta^{2}\left(\frac{\mathfrak{p}^{2}_{I,l}\phi_{I}}{2}-\frac{\mathfrak{p}^{2}_{I,0}\phi_{I}}{2}-\frac{5\mathfrak{p}^{2}_{I,l}\phi_{I}^{3}}{8}+\frac{5\mathfrak{p}^{2}_{I,0}\phi_{I}^{3}}{8}\right)+$
$\displaystyle\mathfrak{p}^{1}_{I,0}\left(\frac{q^{2}}{4}-\frac{5\Delta}{4}+\Delta^{2}\left(\frac{1}{4}-\frac{15\phi_{I}^{2}}{16}\right)\right)+\mathfrak{p}^{1}_{I,l}\left(-\frac{25}{12}-\frac{q^{2}}{4}+\Delta\left(\frac{5}{4}-\frac{25\phi_{I}^{2}}{24}\right)+\Delta^{2}\left(-\frac{1}{4}+\frac{55\phi_{I}^{2}}{48}-\frac{25\phi_{I}^{4}}{192}\right)\right)+$
$\displaystyle\left(\frac{5\mathfrak{p}^{2}_{I,l}\Delta\phi_{I}}{2}+\Delta^{2}\left(-\frac{\mathfrak{p}^{2}_{I,l}\phi_{I}}{2}+\frac{5\mathfrak{p}^{2}_{I,l}\phi_{I}^{3}}{8}\right)+\mathfrak{p}^{1}_{I,l}\left(\frac{q^{2}}{4}-\frac{5\Delta}{4}+\Delta^{2}\left(\frac{1}{4}-\frac{15\phi_{I}^{2}}{16}\right)\right)\right)\log(\rho)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(170) $\displaystyle\mathfrak{p}^{2}=\,$
$\displaystyle\mathfrak{p}^{2}_{I,0}+\mathfrak{p}^{2}_{I,l}\log(\rho)+\rho^{2}\bigg{[}\Delta\left(-\frac{5\mathfrak{p}^{1}_{I,l}\phi_{I}}{12}+\frac{5\mathfrak{p}^{1}_{I,0}\phi_{I}}{12}\right)+\Delta^{2}\left(\frac{\mathfrak{p}^{1}_{I,l}\phi_{I}}{12}-\frac{\mathfrak{p}^{1}_{I,0}\phi_{I}}{12}-\frac{5\mathfrak{p}^{1}_{I,l}\phi_{I}^{3}}{48}+\frac{5\mathfrak{p}^{1}_{I,0}\phi_{I}^{3}}{48}\right)+$
$\displaystyle\mathfrak{p}^{2}_{I,l}\left(-\frac{5}{4}-\frac{q^{2}}{4}-\frac{5\Delta\phi_{I}^{2}}{8}+\Delta^{2}\left(\frac{\phi_{I}^{2}}{8}-\frac{5\phi_{I}^{4}}{64}\right)\right)+\mathfrak{p}^{2}_{I,0}\left(-\frac{5}{6}+\frac{q^{2}}{4}-\frac{5\Delta\phi_{I}^{2}}{12}+\Delta^{2}\left(\frac{\phi_{I}^{2}}{12}-\frac{5\phi_{I}^{4}}{96}\right)\right)+$
$\displaystyle\left(\frac{5\mathfrak{p}^{1}_{I,l}\Delta\phi_{I}}{12}+\Delta^{2}\left(-\frac{\mathfrak{p}^{1}_{I,l}\phi_{I}}{12}+\frac{5\mathfrak{p}^{1}_{I,l}\phi_{I}^{3}}{48}\right)+\mathfrak{p}^{2}_{I,l}\left(-\frac{5}{6}+\frac{q^{2}}{4}-\frac{5\Delta\phi_{I}^{2}}{12}+\Delta^{2}\left(\frac{\phi_{I}^{2}}{12}-\frac{5\phi_{I}^{4}}{96}\right)\right)\right)\log(\rho)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(171)

###  E.2 UV expansions

For the UV expansions we put $A_{U}=0=\chi_{U}$, as explained in Appendix D.
Also, the expansions are considered for $\Delta=5/2$ and are written in terms
of $z\equiv e^{-\rho}$.

The fluctuations of the scalars in the probe approximation are more
complicated to derive in this case as some of the exponents in expansions are
non-rational numbers. The part of the expansion which is of importance in our
calculations will be in the following form:

$\displaystyle\mathfrak{p}^{1}$ $\displaystyle=$
$\displaystyle\log(z)z^{5/2}\mathfrak{p}^{1}_{5/2,l}(1+\cdots)+z^{5/2}\mathfrak{p}^{1}_{5/2}(1+\cdots)\,,$
(172) $\displaystyle\mathfrak{p}^{2}$ $\displaystyle=$ $\displaystyle
z^{\alpha_{-}}\mathfrak{p}^{2}_{-}(1+\cdots)+z^{\alpha_{+}}\mathfrak{p}^{2}_{+}(1+\cdots)\,,$
(173)

where $\alpha_{\pm}\equiv\big{(}5\pm\sqrt{35/3}\big{)}/2$.

##  Appendix F More about the free energy

In this short Appendix, we report some details of the free energy density
study for values of $\Delta$ close to $\Delta\simeq 1.8$. In Fig. 11 we show a
detail of Fig. 8, that highlights the evolution of the free energy of the
confining solutions. We notice that this is monotonic for $\Delta=1.7$, while
it has a local maximum and minimum, and changes in concavity, when
$\Delta=1.95$. It would be tempting to correlate this behaviour with the
appearance of a tachyon in the spectrum, but, as we comment in the main body
of the paper, the presence of divergences, treated via holographic
renormalisation, turns the concavity of ${\cal F}$ into a scheme-dependent
quantity. We hence do not further pursue this line of enquiry in this paper.

Figure 11: The free energy $\hat{\cal F}$ as a function of the source
$\hat{\phi}_{J}$, in units of the scale $\Lambda$, for representative choices
of $\Delta$ near $\Delta\simeq 1.8$, above which the spectrum of fluctuations
contains a tachyon in part of the parameter space. The black curve denotes the
confining solutions, while the red and blue solutions are singular domain-wall
ones.

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