# Tachyonic AdS/QCD, Determining the Strong Running Coupling and
$\beta$-function in both UV and IR Regions of AdS Space

Adamu Issifu<EMAIL_ADDRESS>Departamento de Física, CFM—Universidade
Federal de Santa Catarina,
Caixa Postal 476, 88040-900 Florianópolis, SC, Brazil  Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil  Elijah A. Abbey<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de São Carlos,
Caixa Postal 676, 13565-905 São Carlos, SP, Brazil  Francisco A. Brito
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Campina
Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil
Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970 João Pessoa, Paraíba, Brazil

###### Abstract

In this paper, we investigate QCD-like running coupling
$\alpha_{s}^{AdS}(Q^{2})$ and its associated $\beta$-function $\beta(Q^{2})$
in the spirit of tachyonic AdS/QCD. We distort the $\text{AdS}_{5}$ conformal
symmetry using a color dielectric function $G(\phi(z))$ associated with
tachyons, with $\phi(z)$ the tachyon field. The function presents different
properties of $\alpha_{s}^{AdS}(Q^{2})$ at small and large values of the
fifth-dimensional holographic variable $z$. The function distorts the AdS
space, giving $\alpha_{s}^{AdS}(Q^{2})$ and its $\beta$-function
$\beta(Q^{2})$ at any $Q^{2}$ scale, with $Q^{2}$ the space-like momentum. The
result obtained for a large value of $z$ is expected to show characteristics
similar to nonperturbative QCD. On the other hand, the result obtained for a
small value of $z$ is expected to show characteristics similar to perturbative
QCQ. The presence of free tachyons leads to distortion of the AdS space at a
small $z$, that notwithstanding, condensed tachyon states also lead to large
$z$ distortion. This provides a unified background for determining
$\alpha_{s}^{AdS}(Q^{2})$ and its $\beta(Q^{2})$ in both the ultraviolet (UV)
and infrared (IR) regions using a single function in the framework of
tachyonic AdS/QCD.

##  I Introduction

AdS/CFT correspondence correspondence-a ; correspondence-b  has been receiving
some attention among experts these days. This duality scenario enable us to
establish relation between quantum gravity on $d+1$-dimensional Anti-de Sitter
(AdS) space and $d$-dimensional conformal field theory (CFT). In principle, we
can calculate physical observables in strongly coupled gauge theory in terms
of weakly coupled classical gravity theory. One of the most intriguing
examples correspondence  of the duality; is the correspondence between SU(N)
$\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory with gravity. The
correspondence is the basis for initiating holographic principles by ’t Hooft
correspondence1 . The gravitational dual is type IIB supergravity or string
theory correspondence2 . This correspondence has proven to give a supergravity
explanation to field theories that display confinement and chiral symmetry
breaking characteristics Klebanov ; Maldacena1 . The gauge theory dual to the
AdS space has shown that the potential between two static color point
particles has Coulombic potential contribution to the net potential behavior
Rey ; Maldacena2  indicating deconfinement phase transition. Also,
investigations by (Witten, ) suggest that considering compact boundary, AdS
corresponds to the confinement phase while the black hole resulting from the
AdS phase corresponds to the deconfinement phase. This interesting outcome is
associated with Hawking and Page, transition Hawking , where they considered
two separate solutions to the Einstein equation, the AdS, and the black hole
AdS spaces. Also, QCD phenomenology derived from holographic models has been
applied successfully in studying several strong interacting characteristics
such as; light flavor mesons Bernardini  and glueball phenomenology
Bernardini1  using a more robust approach referred to as the configurational
entropy Gleiser ; Gleiser1 ; Ferreira .

In the AdS/CFT correspondence, the effective tension $T_{\text{string}}$ of
the string and the string coupling $g_{s}$ can be related to the ’t Hooft
coupling $\lambda:=g^{2}N$, as $g_{s}={\lambda}/{4\pi N}$ and
$T_{\text{string}}={\sqrt{\lambda}}/{2\pi}$. Thus, $g_{s}=g^{2}/4\pi$, where
$g$ is the gauge coupling of the SYM theory. In the limit
$N\rightarrow\infty$; $g_{s}\rightarrow 0$, and perturbative string theory is
applicable. On the other hand, in the limit, $\lambda\rightarrow 0$ (weak
gauge coupling), perturbation theory in terms of Feynman diagrams apply to the
gauge theory Hoker ; N Beisert . In the ’t Hooft limit $N\rightarrow\infty$,
$\lambda=\text{fixed}$, spacetime geometry has a curvature, $\mathcal{R}$,
much smaller compared to the string scale $1/g_{s}$, so gravity can be a good
approximation of the gauge theory. Otherwise, both theories are highly
complex, making their equivalence non-trivial. Also, small curvature
$\mathcal{R}$ means large AdS radius $R$, since $\mathcal{R}\sim 1/R^{2}$,
with $R=(4\pi g_{s}N)^{1/4}l_{s}$, where $l_{s}$ is the string length
correspondence . The gauge and the string couplings are related $g_{s}\sim
g^{2}$, and in the ’t Hooft limit, the string coupling becomes small such that
the stringy effect decouples. In this limit, nonplanar interactions are
removed, so we consider free strings on the $\text{AdS}_{5}\times\text{S}^{5}$
background. Additionally, large $N$ is not the same as QCD; however, in many
instances, the results for large $N$ are equivalent to $N_{c}=3$ theories,
with $N_{c}$ QCD color number. The ’t Hooft parameter, $\lambda$, divides
large $N$ QCD into two regions. In the limit $\lambda\ll 1$, we have a
perturbative theory where Feynman diagrams are used to calculate amplitudes.
In the limit $\lambda\gg 1$, we have strong coupling with nonplanar diagrams
Ballon . Also, Light-front holography provides means for establishing a
relationship between the boost-invariant light-front wavefunctions and bound-
state amplitudes in AdS space light-front . Holographic QCD provides a
suitable description of hadrons with known spectroscopic dynamical
characteristics dilaton-profile ; holographic qcd .

In this work, we deform the AdS space Polchinski ; Polchinski1 ; Boschi ;
Boschi1 ; Brodskya ; Brodsky1  with a Higgs-like dimensionless color
dielectric function $G(\phi)$ associated with tachyonic potential (Issifu, ;
Issifu1, ; Issifu2, ; Issifu3, ). The color dielectric function deforms the
AdS space in the UV region in the presence of free tachyons (active tachyons)
whilst tachyon condensed color dielectric function $G(\eta)$, where $\eta$ is
the condensed tachyon field, also deforms the AdS space in the IR region
similar to the positive sign dilaton profile used in determining strong
couplings in AdS/QCD. We show that the tachyon field $\phi$ is associated with
glueball field $\varphi$, and the color dielectric function is also associated
with higher dimensional operator $H_{\mu\nu}H^{\mu\nu}$ coupled to a Standard
Model (SM) gauge field ${\cal O}_{SM}$ that leads to strongly interacting
light glueballs Soni . The AdS action intended for this study will be
formulated from the Dirac-Born-Infeld (DBI) action modified at a tachyonic
vacuum similar to Sen’s $\text{AdS}_{5}$ tachyonic action Ashoke . Firstly, we
investigate the characteristics of both the strong running coupling
$\alpha_{s}(Q^{2})$, and the associated $\beta$-function $\beta(Q^{2})$, by
distorting the AdS space with tachyonic $G(\phi)$. Secondly, we examine
$\alpha_{s}(Q^{2})$ and $\beta(Q^{2})$ by distorting the AdS space with
$G(\eta)$. We expect $\alpha_{s}(Q^{2})$ and $\beta(Q^{2})$ to behave
similarly to the perturbative QCD (pQCD) for the UV deformation of the AdS
space. On the other hand, we expect AdS/QCD or nonperturbative QCD-like
behavior for the IR deformation of the AdS space. The results obtained from
these regions will be compared with effective couplings determined from
different observables, such as; lattice QCD results, QCD phenomenology, and
$g_{1}$ scheme where $\alpha_{g1}(Q^{2})$ is extracted from the well-measured
Bjorken sum rule Adolph ; Deur1 ; Bjorken ; gp1 . This approach will bring a
new perspective to AdS/QCD, where pQCD coupling characteristics can be
determined through direct UV deformation of the AdS space instead of
extrapolation from the IR deformation of the AdS space. Also, we will study
the parameter that controls the transition from the pQCD to nonperturbative
QCD.

Additionally, we show that $\alpha_{s}(Q^{2})$ and $\beta(Q^{2})$ are related
to strongly interacting scalar glueballs with mass $m_{\phi}$ and discuss its
effect. We will discuss any Landau singularity that may be observed in the UV
region and propose how to deal with it in the model framework. Strong running
coupling is a subject of active research due to its limited understanding in
the low momentum transfer region. Good knowledge of $\alpha_{s}(Q^{2})$ at
$Q\,\rightarrow\,\infty$ is necessary to match for the growing precision of
hadron scattering experiments and enhance the understanding of high energy
unification of strongly interacting and electroweak theories. On the other
hand, a precise understanding of $\alpha_{s}(Q^{2})$ at $Q\,\rightarrow\,0$ on
the scale of proton mass enables us to understand hadron structure,
confinement, and hadronization Olive ; Prosperi ; Altarelli ; AdS-book  (and
references therein).

This paper is organized such that in Sec. II we review the well-known
holographic QCD in two subsections: In Sec. II.1 we review the Light-Front
Holography, and in Sec. II.2 we review the AdS/CFT and Holographic QCD. We set
the basis for the study in Sec. III with Strong Interacting Glueballs and
Association With Tachyons. Under this section, we discuss Strong Interacting
Glueballs and SM Particles in Sec. III.1, Confinement at Tachyonic Vacuum in
Sec. III.2 and Tachyonic AdS/QCD Action in Sec. III.3. We proceed to present
Deformation of the AdS Space in the UV Region in Sec. IV under this section;
we discuss the Strong Running Coupling in Sec. IV.1 and the $\beta$-function
in Sec. IV.2. In Sec. V, we study Deformation of the AdS Space in the IR
region, which leads to the study of the associated Strong Running Coupling in
Sec. V.1 and $\beta$-function in Sec. V.2. We finalize by presenting our
Analyses and Conclusion in Sec. VI; with the Analysis in Sec. VI.1, and the
Conclusion in Sec. VI.2.

##  II Review of Holograhic QCD

###  II.1 Light-Front Holography

Light-front (LF) quantization Brodsky2 ; Brodsky2a  provides suitable grounds
for investigating the structure of the hadrons concerning quark and gluon
degrees of freedom. The LF wavefunctions (LFWFs) are the relativistic
generalization of the Schrödinger wavefunctions. They are determined at fixed
time $\tau\,=\,x^{+}\,=\,x^{0}\,+\,x^{3}$ referred to as the LF time Dirac
instead of the ’usual’ fixed time $t$. LF holography range of confinement ;
holographic qcd ; Brodsky3 ; Brodsky4 ; holographic qcd2  relates the
equations of motion in AdS space to QCD Hamiltonian formalism in physical
spacetime quantized on LF at $\tau$. LFWFs in hadron physics are analogous to
Schrödinger wavefunctions in atomic physics Brodsky2 . The correspondence
allows for a direct relationship between hadronic amplitude $\Phi(z)$ in AdS
space and LFWF $\phi_{L}(\zeta)$ describing constituent quarks and gluons that
form the hadron, with $\zeta$ the invariant impact LF variable. The connection
is a consequence of AdS/CFT correspondence, a theory established to be scale
independent correspondence . The term correspondence is applied because we can
map a weakly interacting gravity-related theory in $d+1$-dimensional AdS space
onto any strongly interacting CFT (such as SYM theory) in $d$-dimensions,
giving rise to the name gauge/gravity duality. This mapping generates a
relation between the $5^{th}$-dimension variable ($z$) of AdS space and
$\zeta$ in physical spacetime.

The mapping between $z$ and $\zeta$ was originally obtained by relating the
electromagnetic current matrix element in AdS space Polchinski1  to the
corresponding current matrix element in LF theory holographic qcd ; Abidin .
Also, we can establish a relationship between their energy-momentum matrix
elements Brodsky4 ; this demonstrates the consistency of holographic mapping
to physical observables in LF. The resulting equation for meson bound states
$q\bar{q}$ at $\tau$ has the form of single-variable relativistic Lorentz
invariant Schrödinger equation

$\left(-\dfrac{d^{2}}{d\zeta^{2}}-\dfrac{1-4L^{2}}{4\zeta^{2}}+U(\zeta)\right)\phi_{L}(\zeta)=\mathcal{M}^{2}\phi_{L}(\zeta),$
(1)

where $\mathcal{M}$ is the mass, $L$ relative orbital angular momentum of $q$
(quark) and $\bar{q}$ (antiquark),
$U(\zeta)\,=\,\kappa^{4}\zeta^{2}+2\kappa^{2}(L+S+1)$ is the confining
potential in the soft-wall model with mass-scale $\kappa$ and spin $S$.
Therefore, when the key dynamical impact variable $\zeta$; is correctly
identified, we can reduce the theory to a semi-classical approximation. In
this case, the properties of the strong interacting dynamics; are centered on
an effective potential $U(\zeta)$. It is striking that, in the semi-classical
approximation, the LF Hamiltonian has a structure that exactly agrees with the
eigenvalue equations in the $5$-dimensional AdS space. That permits us to
relate the AdS wave function $\Phi(z)$ to the internal constituent structure
of the hadron. This characteristic yields a relation between the coordinate
$z$ in the AdS space with $\zeta$ Brodsky2 .

###  II.2 AdS/CFT and Holographic QCD

In this section, we review QCD running coupling $\alpha_{s}$ and its
associated $\beta$-function in $\text{AdS}_{5}$ space-time obtained from
conformal field theory. QCD is neither conformal nor supersymmetric, even
though it shows minimal conformal behavior in both at far ultraviolet (UV)
Bjorken1 ; Gross ; Georgi  and deep infrared (IR) regions beta-f-b ; beta-f-b1
with quark-gluon and hadron degrees of freedom respectively. Therefore, to
obtain color confinement, the conformal symmetry in $\text{AdS}_{5}$ space
needs to be broken. This can be accomplished in many different ways. The
general procedure governing the AdS/CFT approach is the isomorphism of the
Poincaré group and conformal transformation SO(4, 2) to the group of
isometrics of $\text{AdS}_{5}$ space, the group then transforms giving us the
$\text{AdS}_{5}$ metric

$ds^{2}=\dfrac{R^{2}}{z^{2}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dz^{2}\right),$
(2)

where $R$ is the AdS radius. The metric is invariant under scale changes in
the fifth dimensional variable $z\,\rightarrow\,\tilde{\lambda}z$ and the
spacetime variable $x^{\mu}\,\rightarrow\,\tilde{\lambda}x^{\mu}$. The $z$
coordinate acts like a scaling variable in the Minkowski space. Different
values of $z$ correspond to a different measure of $4$-momentum for which the
hadron can be studied ads-beta-f . Some well known functions that are
introduced to break the $\text{AdS}_{5}$ space are; dilaton profiles
$e^{\pm\phi_{d}(z)}$, with $\phi_{d}(z)$ the dilaton field, dilaton-profile ;
dilaton-profile1  or a warp factor $e^{A(z)}$, where $A(z)$ is any suitable
function, commonly, logarithmic functions are used i.e.,
$A(z)\,\sim\,\log(z)$. Whichever function is chosen to distort the
$\text{AdS}_{5}$ space leads to confinement in uniquely different form
H-quark-p-ads . A more successful form of this study has been achieved through
LF Holographic mapping to classical gravity when the $\text{AdS}_{5}$ space is
modified by a positive-sign dilaton background. These changes induce an
analytically color confinement leading to nonperturbative AdS/QCD running
coupling, $\alpha^{AdS}_{s}(Q^{2})$. It is important to note that the
understanding of running coupling for pQCD is limited to short distances or
high $Q^{2}$ regimes. In a similar sense, the AdS/QCD with positive dilaton
background is based on long-distance interactions and a small $Q^{2}$ regime
ads-beta-f .

Analytically, the $\text{AdS}_{5}$ action bears resemblance to the general
relativity action $S_{GR}\,\propto\,\int d^{4}x\sqrt{|\det
g_{\mu\nu}|}\tilde{R}/G_{N}$, where $\tilde{R}$ is the Ricci scalar and
$G_{N}$ is the Newton’s constant. In AdS/QCD we replace Ricci scalar with the
gauge field as $\sqrt{\tilde{R}}\rightarrow F^{\mu\nu}$, ${G_{N}}$ is replaced
by the gauge coupling $\sqrt{G_{N}}\rightarrow g_{5}$, and the modulus of the
metric determinant $|\det g_{\mu\nu}|$ is replaced by the modulus of the
$\text{AdS}_{5}$ metric determinant $\sqrt{|\det
g_{\mu\nu}|}\rightarrow\sqrt{|g_{AdS}|}e^{\kappa^{2}z^{2}}$ which includes the
dilaton profile $e^{\kappa^{2}z^{2}}$ to distort the $\text{AdS}_{5}$ space,
here, $\phi_{d}(z)=\kappa^{2}z^{2}$. Therefore, the $\text{AdS}_{5}$ action
takes the form

$S_{AdS}=-\dfrac{1}{4}\int{d^{4}xdz\sqrt{|g_{AdS}|}e^{\kappa^{2}z^{2}}\dfrac{1}{g_{5}^{2}}F^{\mu\nu}F_{\mu\nu}}.$
(3)

From Eq.(2) the determinant of the metric is $\sqrt{|g_{AdS}|}=(R/z)^{5}$ and
from the above expression, after introducing the dilaton profile, it turns
into $\sqrt{|g_{AdS}|}=(R/z)^{5}e^{5\kappa^{2}z^{2}}$. The prefactor

$g_{5}^{-2}(z)=g_{5}^{-2}e^{\kappa^{2}z^{2}},$ (4)

restores the conformal symmetry in Eq.(3) and leads to color confining
properties, in this case, $g_{5}^{2}$ has length dimension. Now, comparing
$g_{5}$ to the coupling in the $\mathcal{N}\,=\,4$ SYM theory $g$, and mapping
$z\,\rightarrow\,\zeta$, we can calculate the AdS/QCD
$\alpha_{s}^{AdS}(Q^{2})$ and the associated $\beta(Q^{2})$ AdS-book .

##  III Strong interacting Glueballs and Association With Tachyons

In this section, we intend to establish the relationship between the hidden
scalar glueball state and the tachyonic field used in this paper. Glueballs
are candidates for dark matter particles with significant implications in
cosmology and particle physics. Also, some glueballs may be self-interacting
through strong force with crucial astrophysical implications Soni ; Boddy ;
Dietrich ; Appelquist . These particles may be observed in particle physics
experiments when established that they interact with SM particles through high
dimensional operators. We will establish a relationship between glueballs and
tachyons and calculate the QCD-like strong running coupling $\alpha_{s}$ and
the associated $\beta$-function in a QCD-like fashion using Higgs-like
tachyonic potential.

###  III.1 Strong Interacting Glueballs and SM Particles

It has been demonstrated that glueballs interact with the SM sector through
high dimensional operators, as will be discussed below. That means glueballs
may be observed through particle physics experiments instead of the over-
reliance on gravitational means that have proven difficult over the years. In
Ref. Soni  the authors investigated this phenomenon using a simple setup to
establish the interaction between the glueballs and the SM particles through
higher dimensional operators. In their model, they derived

$\mathcal{L}=\dfrac{1}{M^{n}}H_{\mu\nu}H^{\mu\nu}{\cal O}_{SM},$ (5)

where $M$ is the cutoff scale, $n$ is a positive constant related to the
chosen spacetime dimension, and the choice of the SM interaction term ${\cal
O}_{SM}$ is open. Consequently, we can express

$\mathcal{L}=\dfrac{1}{M^{4}}H_{\mu\nu}H^{\mu\nu}(F_{\alpha\beta}F^{\alpha\beta})\;\rightarrow\;\dfrac{Nm^{3}}{M^{4}}\varphi
F_{\alpha\beta}F^{\alpha\beta},$ (6)

where $F_{\alpha\beta}$ is the abelian field strength, $m$ is the glueball
mass, $H_{\mu\nu}H^{\mu\nu}$ is the higher dimensional operator, $N$ is
unspecified number of colors from $\text{SU}(N)$ gauge group and $\varphi$ is
the lightest glueball field. Also, $N$ is chosen to make $\varphi$ self-
interacting. Detailed discussion on this subject is contained in Soni  and the
references therein.

###  III.2 Confinement at Tachyonic Vacuum

To derive an equivalence relation for the tachyonic fields for comparison with
Eq.(6); we start with the Dirac-Born-Infeld (DBI) action Born ; Gibbons ;
Dirac1 ; Polchinskia ; Taylor  and subsequently, we will modify it with the
tachyonic field at tachyonic vacuum. The interesting feature about bosonic
opened strings at tachyonic vacuum is that they behave like closed strings
with no D$p$-branes. In that view, the endpoints of the opened strings are
connected by a flux line on the D$p$-brane worldvolume to form closed strings
Sent1 ; Sen1 ; Recknagel ; Sen2 ; Harvey ; Kostelecky ; Sen ; Zwiebach2 ;
Kostelecky1 ; Berkovits ; Harvey1 ; Gopakumar ; Gerasimov . Also, in this
region, the negative tachyon energy density $V(\phi)$ exactly cancels the
D$p$-brane tension $\varepsilon_{p}$, i.e.,
$V(\phi)\,+\,\varepsilon_{p}\,=\,0$ Sen ; Zwiebach2 ; Kostelecky ; Berkovits ;
Witten2 . Therefore, there is no energy cost in rearranging the strings on the
remaining lower-dimensional branes in the region, and there are no opened
string excitations as well Sent2 ; Bigazzi . Thus, the remaining lower-
dimensional branes connecting the endpoints of the strings; serve as the
source and sink of the color charges. In the current study, the particles
involved are valence gluons. As a result, the closed strings formed at the
tachyonic vacuum; are treated in the spirit of color flux-tube picture in
chromoelectric flux confinement associated with QCD theory. Here, the opened
stings considered are initially living on nonperturbative D$p$-branes Leigh .
That makes them suitable for applications in nonperturbative QCD. We will
focus on the gauge field for the current analyses. Thus all the other fields
are decoupled for convenience. Therefore, the DBI action takes the form

$S_{\text{DBI}}=-T_{p}\int d^{p+1}\xi
e^{-\phi_{d}}\sqrt{-\det(\eta_{\mu\nu}+(2\pi\alpha^{\prime})F_{\mu\nu})},$ (7)

where $F_{\mu\nu}\,=\,\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
abelian gauge field strength living on the worldvolume of the D$p$-brane,
D$p$-brane tension

$\tau_{p}=\dfrac{T_{p}}{g_{s}}=\dfrac{1}{g_{s}\sqrt{\alpha^{\prime}}(2\pi\sqrt{\alpha^{\prime}})^{p}},$
(8)

string tension

$T_{\text{string}}=\dfrac{1}{2\pi\alpha^{\prime}},$ (9)

and string coupling $g_{s}=e^{\langle\phi_{d}\rangle}$. To understand the
strong interaction dynamics of the opened strings on the D$p$-brane
worldvolume, we derive the dimensional reduced $\text{U}(1)$ Yang-Mills theory
Taylor1 ; Banks  by small perturbation in Eq.(7), which yields

$S_{YM}=-\tau_{p}V_{p}-\dfrac{1}{4g_{YM}^{2}}\int d^{p+1}\xi
F_{\mu\nu}F^{\mu\nu}+{\cal O}(F^{4}).$ (10)

Here, $V_{p}$ is the $p$-brane worldvolume, which can be ignored subsequently,
and $g_{YM}$ is the Yang-Mills coupling given by,

$g_{YM}^{2}=\dfrac{1}{(2\pi\alpha^{\prime})^{2}\tau_{p}}=\dfrac{g_{s}}{\sqrt{\alpha^{\prime}}}\left(2\pi\sqrt{\alpha^{\prime}}\right)^{p-2},$
(11)

hence, the relation $g_{YM}^{2}\,\sim\,g_{s}$, that forms the basis for the
gauge/gravity duality; is established. This expression reduces to the $10\,d$
$\mathcal{N}\,=\,1$ Super Yang-Mills action. Moreover, since the mass of the
string is proportional to its length, the closer the branes, the less massive
the attached strings may be Witten1 ; Taylor . Thus, for stacks of $N$
D$p$-branes, we have $\text{U}(1)^{N}$ gauge groups on its worldvolume which
can be decomposed as $\text{U}(N)=\text{SU}(N)\times\text{U}(1)$. That can
further be decomposed into standard model symmetry depending on the number of
D$p$-branes crossing each other Lust . Now we modify Eq.(7) with a tachyon
potential $V(\phi)$ Garousi , with $\phi$ the tachyon field

$\displaystyle S$ $\displaystyle=-T_{p}\int d^{p+1}\xi
e^{-\phi_{d}}V(\phi)\sqrt{-\det(\eta_{\mu\nu}+(2\pi\alpha^{\prime})F_{\mu\nu})}$
$\displaystyle=-\tau_{p}\int d^{p+1}\xi
V(\phi)\left[1+\dfrac{1}{4}(2\pi\alpha^{\prime})^{2}F_{\mu\nu}F^{\mu\nu}+{\cal
O}(F^{4})\right],$ (12)

in the last step, we considered slowly varying tachyons with a constant
dilaton field $\langle\phi_{d}\rangle$. At the minimum of the potential
$V(\phi_{0})$ the action disappears Sent2 ; Sent1 ; Recknagel . Also, the
potential is symmetric under $\phi_{0}\,\rightarrow\,-\phi$, with maximum at
$\phi\,=\,0$ and minimum at $\phi_{0}\,=\,\pm a$. We will disregard the first
term and consider only the gauge fields propagating on the D$p$ worldvolume
for the analysis because the gauge field is enough for the current study.
Additionally, we introduce a dimensionless function
$G(\phi)\,=\,(2\pi\alpha^{\prime})^{2}V(\phi)$, which will be referred to as
the color dielectric function that drives the confinement and deconfinement
transitions. Consequently,

$\mathcal{L}=-\dfrac{1}{4g_{s}}G(\phi)F_{\mu\nu}F^{\mu\nu},$ (13)

for $T_{p}\,=\,1$. Comparing Eqs.(6) and (13), we can identify the higher
dimensional operator, $H_{\mu\nu}H^{\mu\nu}$ with $G(\phi)$ and the glueball
field $\varphi$ with the tachyonic field $\phi$. Introducing the kinetic
energy component of the tachyon field into the last step of Eq.(III.2), we
studied the phenomenology of confinement in references Issifu ; Issifu1 ;
Issifu2 ; Issifu3 ; dielectric-f .

###  III.3 Tachyonic AdS/QCD Action

Considering $5$D holographic QCD models, the background flavor branes are
D$4$-$\bar{\text{D}}4$ system Bigazzi1 ; Gursoy . However, in this study, we
consider opened strings with their endpoints on the same brane, so we consider
a background D$4$-brane. Hence, we can express $\text{AdS}_{5}$ action similar
to the one proposed in Ashoke  adopted in Iatrakis  for studying hadronic
properties in the light of AdS/QCD, from Eq.(III.2)

$S=-\dfrac{1}{4g_{5}^{2}}\int d^{4}xdzG(\phi(z))F^{\mu\nu}F_{\mu\nu},$ (14)

where $G(\phi(z))$ is only a function of $z$ and
$g_{s}\,\rightarrow\,g_{5}^{2}$ has the dimension of length. Additionally,
tachyonic corrections were introduced in Ref. Barbosa  to show equivalence in
AdS/QCD and tachyonic AdS/QCD Bajnok  in $4$D holographic dual to $5$D weakly
coupled gravity using similar action. The $G(\phi)$ will be carefully defined
to achieve the phenomenon of confinement and asymptotic freedom in separate
energy regimes. Again, we demonstrate that the tachyonic color dielectric
function, $G(\phi)$, deforms the $\text{AdS}_{5}$ metric at the asymptotic
$\text{AdS}_{5}$ bulk in the UV limit ($z\,\rightarrow\,0$) corresponding to
deconfinement transition associated with $\text{AdS}_{5}$ black holes Hawking
. On the other hand, deforming the $\text{AdS}_{5}$ metric with condensed
tachyonic color dielectric function $G(\eta)$ leads to IR deformation of the
$\text{AdS}_{5}$ bulk leading to the phenomenon of color confinement in the IR
regime ($z\,\rightarrow\,\infty$), similar to deforming the ’usual’
$\text{AdS}_{5}$ space with positive dilaton background Brodsky1 . Here, the
confining transition corresponds to color singlet glueball states with $N^{0}$
degree of freedom whilst the deconfinement transition corresponds to the state
where the gluons are free with $N^{2}$ degrees of freedom Witten ; Ballon .

##  IV Deformation of the AdS Space in the UV Region

###  IV.1 Strong Running Coupling

The strong running coupling $\alpha_{s}^{AdS}(Q^{2})$ can be determined by
defining a new coupling $g_{5}(z)$ with dependence on $z$, that restores the
conformal symmetry in Eq.(14),

$g_{5}^{-2}(z)=g_{5}^{-2}G(\phi(z)).$ (15)

The deformation of the conformal symmetry in Eq.(14) is relevant for obtaining
running coupling and a nonzero $\beta$-function because conformal symmetry
leads to constant coupling and zero $\beta$-function. The scale-dependent
gauge field depicted by the expression above should be regarded as a
consequence of gauge field strength renormalization,
$F^{\mu\nu}\,\rightarrow\,G^{1/2}F^{\mu\nu}$. In ’regular’ QCD theory, we can
rescale the gluon field as $A^{\mu}\,\rightarrow\,\tilde{\lambda}A^{\mu}$
which leads to $G^{\mu\nu}\,\rightarrow\,\tilde{\lambda}G^{\mu\nu}$, with
$G^{\mu\nu}$ the non-abelian field strength, in the QCD Lagrangian density,
$\mathcal{L}_{\text{QCD}}$, this is compensated with rescaling the coupling
strength $g\,\rightarrow\,\tilde{\lambda}^{-1}g$. Thus, the renormalization of
the physical QCD coupling $g_{phys}\,=\,Z^{1/2}_{3}g_{0}$ Nayak , with $Z_{3}$
the gluon propagator renomalization factor and $g_{0}$ the bare coupling in
UV-regulated theory similar to renormalization of gauge field and gauge field
strength: $A_{ren}^{\mu}\,=\,Z^{-1/2}_{3}A_{0}^{\mu}$,
$G^{\mu\nu}_{ren}\,=\,Z_{3}^{-1/2}G_{0}^{\mu\nu}$ resulting into a rescaled
Lagrangian
$\mathcal{L}_{\text{QCD}}^{ren}\,=\,Z_{3}^{-1}\mathcal{L}_{\text{QCD}}^{0}\,=\,(g_{phys}/g_{0})^{-2}\mathcal{L}_{\text{QCD}}^{0}$.
All quantities with subscript or superscript ’$0$’ are bare and the ones with
’ren’ are their renormalized counterparts.

Moreover, we can relate the length scale $z$ with the invariant LF impact
variable $\zeta$, which appears in the LF Hamiltonian. The nonconformal
dynamics, confinement and deconfinement transitions associated with QCD will
now be investigated through $g_{5}(z)$ where,
$g_{5}(z)\,\rightarrow\,g_{YM}(\zeta)$ — see Eq.(10) for the comparison with
YM theory. Therefore, we define $\text{AdS}_{5}$ coupling constant,
$\alpha^{AdS}_{s}\,\equiv\,g_{5}^{2}/4\pi$, which requires $Q^{2}$-dependence
to fully describe the dynamics of the particles in both UV and the IR domains,
corresponding to short and long-distance interparticle separations,
respectively. What makes the $G(\phi)$ interesting is that it distorts the
$\text{AdS}_{5}$ space curvature in two different forms. One in the UV regime
and the other in the IR regime, so perturbation and nonperturbative
characteristics of the QCD-like coupling can be investigated in a single
framework ads-beta-f . We can relate the LF strong coupling with
$G(\phi(\zeta))$ as

$\alpha_{s}^{AdS}(\zeta)=\dfrac{g^{2}_{YM}}{4\pi}\propto G(\phi(\zeta))^{-1}.$
(16)

It is necessary to identify the nature of $G(\phi(\zeta))$ in order to compute
$\alpha^{AdS}_{s}(Q^{2})$ and $\beta(Q^{2})$. The form of $G(\phi(\zeta))$
have been proposed in Refs. Issifu ; Issifu1 ; Issifu2 ; Issifu3  and used in
investigating phenomenon of (de)confinement of glueballs and glueball-fermion
mix states in $1+3$-dimensions. As we have demonstrated in Sec. III.2, the
dimensionless $G(\phi(\zeta))$ is directly associated with the tachyon
potential $V(\phi)$ for strings with $T_{\text{string}}\,=\,1$ corresponding
to $(2\pi\alpha^{\prime})^{2}\,=\,1$, thus, we adopt

$V(\phi(\zeta))=G(\phi(\zeta))=\dfrac{\Lambda}{4}[\phi(\zeta)^{2}-a^{2}]^{2}.$
(17)

In the context of the tachyon potential; $a$ and $\phi$ are dimensionful
constants with energy dimension and $\Lambda$ is a dimensionless constant.
However, in the context of the $G(\phi)$; these quantities are dimensionless
due to $(2\pi\alpha^{\prime})^{2}$ multiplying the potential. We use a linear
ansatz $\phi(z)\,=\,\alpha z$, where $\alpha$ is the tachyon decay scale and
$f_{\alpha}\,=\,1/\alpha$ is the decay constant, hence,

$G(\phi)=\dfrac{\Lambda}{4}[(\zeta\alpha)^{2}-a^{2}]^{2},$ (18)

where $\alpha$ is a dimensionful constant with the dimension of energy. The
scale anomaly in QCD induces a nonvanishing energy-momentum trace tensor,
given as

$\left\langle\Theta^{\mu}_{\mu}\right\rangle=-\dfrac{9}{8}\left<\dfrac{\alpha_{s}}{\pi}G^{\mu\nu}G_{\mu\nu}\right>,$
(19)

where $\alpha_{s}$ is the strong coupling and $G^{\mu\nu}$ is the non-abelian
gauge field strength. Using parton-hadron duality principle Schechter ; Carter
; Campbell ; Kochelev1

$\displaystyle\left\langle\Theta^{\mu}_{\mu}\right\rangle$
$\displaystyle=\left(4V(\phi)-\phi\dfrac{dV}{d\phi}\right)\Big{|}_{\phi_{0}=0}$
$\displaystyle=4V(\phi_{0})=\Lambda a^{4},$ (20)

substituting Eq.(18) into the above expression, we obtain the Bag constant,
$-B_{0}\,=\,\Lambda a^{4}$. Equating the Bag constant to the gluon condensate
yields,

$\Lambda=\dfrac{9\left<\alpha_{s}G^{\mu\nu}G_{\mu\nu}\right>}{8\pi a^{4}},$
(21)

hence, the coupling $\Lambda$, is proportional to the gluon condensate
Kharzeev ; Ellis . Calculating the physical running coupling measured on
$Q^{2}$-scale, we use Mellin transform,

$\displaystyle\alpha^{AdS}_{s}(Q^{2})$
$\displaystyle\sim\int^{\infty}_{0}{\alpha^{AdS}_{s}(\zeta)\zeta^{Q^{2}/m_{\phi}^{2}-1}d\zeta}$
$\displaystyle=\dfrac{\sigma\pi}{\Lambda
a^{4}m^{2}_{\phi}}\left(-\dfrac{\alpha^{2}}{a^{2}}\right)^{-Q^{2}/2m_{\phi}^{2}}(2m_{\phi}^{2}-Q^{2})\csc\left(\dfrac{\pi
Q^{2}}{2m_{\phi}^{2}}\right)$
$\displaystyle=\dfrac{\sigma\Lambda\pi}{m_{\phi}^{6}}\left(\dfrac{m_{\phi}^{2}}{\Lambda}f_{\alpha}^{2}\right)^{Q^{2}/2m_{\phi}^{2}}(2m_{\phi}^{2}-Q^{2})\csc\left(\dfrac{\pi
Q^{2}}{2m_{\phi}^{2}}\right).$ (22)

We substituted Eq.(16), and introduced $m^{2}_{\phi}$, the glueball mass into
the integral to obtain the above expression. We will demonstrate below that,
the glueball mass is related to the scale $Q_{\Lambda}$ at which pQCD breaks
down in the model framework. Also, $\sigma$ is a dimensionful proportionality
constant associated with the Regge slope,
$\sigma\,\sim\,1/(2\pi\alpha^{\prime})^{2}$ with dimension,
$[\sigma]\,=\,\text{energy}^{4}$. We showed in Eq.(III.2) and stated below it
that, $G(\phi)$ is a dimensionless function associated with the tachyon
potential through the Regge slope, so the proportionality constant serves as a
dimensional correction to the equation since dimensionful quantities such as
the glueball mass are introduced. It should be noted that in the tachyon
potential $a$ and $\phi$ have dimensions of energy but in $G(\phi)$ these
quantities are dimensionless. Hence, the proportionality constant $\sigma$ is
introduced to make $\alpha$ and $a$ dimensionful. In the last step, we
substituted
$-m_{\phi}^{2}\,=\,V^{\prime\prime}(\phi)|_{\phi_{0}=0}=a^{2}\Lambda$,
obtained from Eq.(17) around its ’false’ vacuum; here, $m_{\phi}^{2}\,>\,0$
and $a^{2}\Lambda\,<\,0$. For mathematical convenience, we set the
dimensionless coupling constant, $\Lambda$, to unity $\Lambda\,=\,1$, and also
recall that $(2\pi\alpha^{\prime})^{2}$ has also been set to unity below
Eq.(III.2), consequently; $\sigma\,\sim\,1$, but we shall keep it for
dimensional consistency. Therefore, Eq.(IV.1) simplifies to

$\alpha^{AdS}_{s}(Q^{2})=\dfrac{\sigma\pi}{m_{\phi}^{6}}\left({m_{\phi}^{2}f_{\alpha}^{2}}\right)^{Q^{2}/2m_{\phi}^{2}}(2m_{\phi}^{2}-Q^{2})\csc\left(\dfrac{\pi
Q^{2}}{2m_{\phi}^{2}}\right).$ (23)

The negative mass square represents instability in this region due to free
tachyons. Since glueballs are bond states of pure gluons with gluon degrees of
freedom, and $\Lambda$ is proportional to the gluon condensate, negative
$m_{\phi}^{2}$ implies a change in the sign of the gluon condensate, a
phenomenon expected in the quark-gluon-plasma (QGP) regime. It has been shown
in Ref.Issifu2  that in the perturbative UV region, the square of the glueball
mass $m^{2}_{\phi}$ changes sign, i.e.,
$m_{\phi}^{2}\rightarrow\,-m_{\phi}^{2}$. These glueball masses’ presence
reflects the QGP’s nonperturbative properties beyond the transition
temperature $T\,>\,T_{C}$: Indeed, the isoscalar glueball mass, and the gluon
condensate, are expected to change sign at $T\,>\,T_{C}$ Kochelev ; Mathieu .
Lattice QCD calculations for pure $\text{SU}(3)_{c}$ affirms this sign changes
Miller . Additionally, since the four-momentum of tachyons is necessarily
space-like due to their imaginary mass, they can be seen with negative energy
in some inertial frames. In this case, we can reverse the sign of the time
intervals. Thus, negative tachyon energy moving backward in time can be
thought of as a positive energy tachyon moving forward in time. As a result,
the sign of the tachyon decay constant $f_{\alpha}$, which can be associated
with glueball decay, can be altered $f_{\alpha}\,\rightarrow\,-f_{\alpha}$
without affecting the physical properties of the particles Bose ; Dhar .

(a) Left Panel (b) Right Panel

Figure 1: A graph of $\alpha^{AdS}_{s}(Q^{2})/\pi$ against $Q$ for
$m_{\phi}\,=\,0.86\,\text{GeV}$ (left) and $m_{\phi}\rightarrow\infty$
(right).

We compare the graph (left) with the ones contained in Refs. Brodskya ; AdS-
book  for $\alpha_{g1}(Q)/\pi$ (pQCD) and other experimental data in Deur ,
and we observe the expected decrease of $\alpha_{s}$ at large $Q$. We present
results for both; the lightest glueball mass (left) and the heaviest possible
glueball mass (right). However, the theory governing the study is consistent
with light glueball masses, so the graph in the right panel is for analytical
purposes and may not have any physical consequences in the current study. We
observe that $\alpha_{s}^{AdS}(Q^{2})$ decreases sharply with increasing $Q$;
on the other hand, an increase in $m_{\phi}\,\rightarrow\,\infty$ moves the
graph more towards a small $Q$ region beyond the Landau pole, $Q_{\Lambda}$,
(vertical blue line) and also falls quickly at large $Q$. Thus, heavy
glueballs are more likely to be confined than lighter ones.

We plot the leading order of the perturbative result of the QCD running
coupling given by

$\alpha_{s}^{pQCD}(Q^{2})=\dfrac{4\pi}{\beta_{0}\ln\left(\dfrac{Q^{2}}{\Lambda^{2}_{QCD}}\right)}$
(24)

where $\Lambda_{QCD}$ is the QCD scale,

$\beta_{0}=11-\dfrac{2}{3}n_{f},$ (25)

is the first term of the $\beta$-series, with $n_{f}$ the number of quark
flavors active at scale $Q^{2}$ AdS-book  for comparison with our results.

Figure 2: A graph of $\alpha_{s}^{pQCD}(Q^{2})/\pi$ (blue dashed) and
$\alpha^{AdS}_{s}(Q^{2})/\pi$ (red solid line) against $Q$ for
$\Lambda_{QCD}\,=\,0.58\,\text{GeV}$.

We set $n_{f}\,=\,0$ in this plot to consider pure gluon states for the
comparison. The dashed vertical blue line and the faint solid blue line
superimposed on each other are the positions of the Landau pole for both
$\alpha_{s}^{pQCD}(Q^{2})$ and $\alpha_{s}^{AdS}(Q^{2})$ respectively. We
chose the position of the Landau poles for $\alpha_{s}^{pQCD}(Q^{2})$ and
$\alpha_{s}^{AdS}(Q^{2})$ to coincide with each other for easy comparison.

###  IV.2 $\beta$-Function

The $\beta$-function is calculated using the renormalization group theory,

$\displaystyle\beta(\alpha^{AdS}(Q^{2}))$
$\displaystyle=Q^{2}\dfrac{d\alpha^{AdS}(Q^{2})}{dQ^{2}}$
$\displaystyle=-\dfrac{Q^{2}\pi\sigma}{2m_{\phi}^{8}}\left({f^{2}_{\alpha}m^{2}_{\phi}}\right)^{Q^{2}/2m^{2}_{\phi}}\csc\left(\dfrac{\pi
Q^{2}}{2m^{2}_{\phi}}\right)\left\\{(2m^{2}_{\phi}-Q^{2})\left[\pi\cot\left(\dfrac{\pi
Q^{2}}{2m^{2}_{\phi}}\right)-\log[f_{\alpha}^{2}m^{2}_{\phi}]\right]+2m^{2}_{\phi}\right\\}.$
(26)

The ”Principle of Conformality” is satisfied, i.e.,

$\beta(Q\rightarrow 0)=\beta(Q\rightarrow\infty)=0.$ (27)

In the classical approximation to QCD without hadronization, $\beta$-function
vanishes, and the corresponding theory is conformal beta-f1 . This behavior is
maintained when quantum corrections are included for large $Q^{2}$ due to
asymptotic freedom and attain a finite value close to $Q\,\rightarrow\,0$ due
to infrared fixed point. Infrared fixed point translates naturally into color
confinement beta-f2  with hadron degrees of freedom. Because the propagator of
the colored particles has their maximum wavelength in the IR regime, all the
loop integrals for the self-interacting gluon energy decouples at
$Q\,\rightarrow\,0$ beta-f3 . The other condition is,

$\beta(Q)<0\qquad\text{for}\qquad Q>0,$ (28)

where $Q_{\Lambda}$ is the point at which pQCD fails. This condition reflects
the anti-screening behavior of QCD theory at large $Q$ where its running
coupling, $\alpha_{s}$, vanishes due to asymptotic freedom with quark-gluon
degrees of freedom. The QCD $\beta$-function is generally negative, and its
value increases from the UV regime and assumes its maximum value in the IR
domain.

(a) Left Panel (b) Right Panel

Figure 3: A graph of $\beta(Q^{2})$ against $Q$ for
$m_{\phi}\,=\,0.86\,\text{GeV}$ (left) and $m_{\phi}\rightarrow\infty$
(right).

The graph (left) is compared with $\beta(Q)$ (pQCD) reproduced in Refs.
Brodskya ; ads-beta-f . The graph in the right panel shows the behaviour of
the $\beta(Q^{2})$ for $m\,\rightarrow\,\infty$.

We expect the $\beta$-function to have it usual analytical behavior in both
the UV and the IR regimes with or without the Landau pole in $\alpha_{s}$. The
’usual’ behavior of $\beta(Q^{2})$ is due to its conformal invariant
characteristics leading to the well-behaved cut-off in both the IR and the UV
regimes. Regardless of the singularity exhibited by $\alpha_{s}$.

The result in Eq.(23) displays the expected decrease in $\alpha_{s}$ at large
$Q^{2}$ similar to the running coupling displayed for pQCD in Fig. 2. We
determine the position of the Landau pole $Q_{\Lambda}$ in Eq.(23) using the
relation for determining the transition point,

$\dfrac{d\beta}{dQ}=0.$ (29)

However, the results obtained from the above operation are not analytically
solvable for $Q_{\Lambda}$ because of the trigonometric functions involved. So
we expand it for small $Q$ leading to

$\dfrac{8}{m^{2}_{\phi}Q^{3}}+\dfrac{\left(\pi^{2}-6\log[f_{\alpha}^{2}m^{2}_{\phi}]+3\log[f_{\alpha}^{2}m^{2}_{\phi}]\right)Q}{3m_{\phi}^{6}}+{\cal
O}(Q)^{3}=0,$ (30)

this result yields,

$Q_{\Lambda}=\left(\dfrac{24}{-\pi^{2}+6\log[f_{\alpha}^{2}m^{2}_{\phi}]-3\log[f_{\alpha}^{2}m^{2}_{\phi}]}\right)^{1/4}m_{\phi}.$
(31)

Using the same parameters used in plotting Figs. 1, 2 and 3, i.e.,
$m_{\phi}\,=\,0.86\,\text{GeV}$ and $f_{\alpha}\,=\,0.50\,\text{GeV}^{-1}$ we
obtain $Q_{\Lambda}\,\approx\,0.58\,\text{GeV}$; as the position of the Landau
pole. The position of the pole is denoted by a solid blue line in Figs. 1, 2
and 3 above. The Landau singularity signals the scale at which pQCD breaks
down hence, it is often related to the confinement scale or hadronic mass
scale. The singularity is also attributed to the scale dependence of
$\alpha_{s}$. Hence, it serves as an expansion parameter for other quantities
coupling  such as the $\beta$-function.

##  V Deformation of the AdS space in the IR regime

For nonperturbative glueballs, we perturb the tachyon fields about its ’true’
vacuum $\phi_{0}\,=\,\pm a$ by introducing a new field $\eta$ through
$\phi\,\rightarrow\,\eta\,+\,\phi_{0}$. This stabilizes the tachyons and the
new field acquires a positive mass $V^{\prime\prime}(\phi)|_{\phi_{0}=\pm
a}\,=\,2a^{2}\lambda\,=\,m_{\phi}^{2}$ consequently, Eq.(17) become

$\displaystyle V(\eta)$
$\displaystyle=V(\phi)\big{|}_{\phi_{0}}+V^{\prime}(\phi)\big{|}_{\phi_{0}}\eta+\dfrac{1}{2}V^{\prime\prime}(\phi)\big{|}_{\phi_{0}}\eta^{2}$
$\displaystyle=\dfrac{1}{2}m_{\phi}^{2}\eta^{2},$ (32)

this process coincides with tachyon condensation. The new field $\eta$ is
identified with a stable glueball field Issifu3 ; Issifu2  in the IR regime
with a solution

$\eta(\zeta)=\dfrac{e^{m_{\phi}\zeta}}{\zeta},$ (33)

which produces confinement of glueballs in the limit of low energy or large
particle separation distance where color confinement is expected,
$\zeta\,\rightarrow\,\zeta_{*}$ Issifu . Here, $\zeta_{*}$ is the separation
distance within which confinement occurs. The choice of Eq.(33) was selected,
guided by the introduction of the kinetic energy term of $\phi$ into
Eq.(III.2) and solving the resulting equations of motion in the static sector
for the new field $\eta$ in spherical coordinates that leads to
phenomenological Cornell-like confining potential for heavy quarks in $3$D. We
have carried out this studies comprehensively in our previous papers Issifu ;
Issifu1 ; Issifu2 ; Issifu3 , and established that in the limit of large
spatial separation distance $r\,\rightarrow\,r_{*}$ the solution Eq.(33) is
dominant.

###  V.1 Strong Running Coupling

Using the color dielectric function for the condensed tachyons, Eq.(16) alters
to

$\alpha_{s}^{AdS}(\zeta)\propto G(\eta(\zeta))^{-1}.$ (34)

Using Laplace transform

$\displaystyle\alpha_{s}^{AdS}(Q^{2})$
$\displaystyle\sim\int_{0}^{\infty}\alpha_{s}^{AdS}(\zeta)e^{-Q^{2}\zeta/m_{\phi}}d\zeta$
$\displaystyle=\dfrac{4m_{\phi}\rho}{(2m_{\phi}^{2}+Q^{2})^{3}},$ (35)

where $\rho$ is a proportionality constant. At $Q\,=\,0$,
$\rho\,=\,2m_{\phi}^{5}\alpha_{s}^{AdS}(0)$, and the above expression
transforms into

$\alpha_{s}^{AdS}(Q^{2})=\dfrac{(2m_{\phi}^{2})^{3}}{(2m_{\phi}^{2}+Q^{2})^{3}}\alpha^{AdS}_{s}(0).$
(36)

(a) Left Panel (b) Right Panel

Figure 4: A graph of $\alpha_{s}^{AdS}(Q^{2})/\pi$ against $Q$ for
$m_{\phi}\,=\,1.73\,\text{GeV}$ (left) and $m_{\phi}\,\rightarrow\,\infty$
(right).

The graph in the left panel has a similar behavior with the effective coupling
for $\alpha_{s,g1}(Q)$ in Refs. Brodskya ; AdS-book ; ads-beta-f  normalized
at $\alpha_{s}^{AdS}(0)\,=\,\pi$. In the right panel, we observe that
increasing $m_{\phi}\,\rightarrow\,\infty$ reduces the deconfinement energy
threshold and strengthens confinement.

###  V.2 $\beta$-Function

We can also calculate the $\beta$-function from Eq.(IV.2) which leads to the
expression

$\beta(Q^{2})=-\dfrac{3(2m_{\phi}^{2})^{3}Q^{2}}{(2m_{\phi}^{2}+Q^{2})^{4}}\alpha^{AdS}_{s}(0).$
(37)

This expression satisfies the restrictions

$\beta(Q\rightarrow
0)=\beta(Q\rightarrow\infty)=0\qquad\text{and}\qquad\beta(Q)<0\qquad\text{for}\qquad
Q>0.$ (38)

The first condition satisfies the restrictions that QCD approximates conformal
theory in both the far UV and the deep IR regimes, known as ”the principle of
maximum conformality”: meanwhile, the second condition is a consequence of the
asymptotic freedom properties of QCD theory. Hence, $\alpha_{s}^{AdS}(Q^{2})$
can be normalized to $\alpha_{s,g_{1}}$ scheme. A well-measured effective
charge obtained from the Bjorken sum rule Bjorken , for polarized deep
inelastic lepton-proton scattering, by exploring the similarities in the
behavior of the $\beta$-functions. It vanishes in the deep infrared domain,
rises to higher negative intermediately, and fall-off to zero in the far UV
domain. We take advantage of these similar features and normalize
$\alpha_{s}^{AdS}(Q^{2}=0)=\pi$ for comparison. Taking the derivative with
respect to $Q$,

$\dfrac{d\beta}{dQ}=-\dfrac{6(2m_{\phi}^{2})^{3}Q\pi}{(2m^{2}_{\phi}+Q^{2})^{4}}\left[1-\dfrac{4Q^{2}}{2m^{2}_{\phi}+Q^{2}}\right],$
(39)

consequently, there is a transition at $Q_{0}\,=\,\sqrt{(2/3)}m_{\phi}$.
Perturbation theory is applicable in the regime
$Q_{0}\,>\,\sqrt{(2/3)}m_{\phi}$ whilst non-perturbation theory is applicable
in the regime $Q_{0}\,<\,\sqrt{(2/3)}m_{\phi}$. Also, the conditions

$\dfrac{d\beta}{dQ}\Big{|}_{Q=Q_{0}}=0,\qquad\qquad\dfrac{d\beta}{dQ}\Big{|}_{Q_{0}}<0\quad\text{for}\quad
Q<Q_{0}\qquad\text{and}\qquad\dfrac{d\beta}{dQ}\Big{|}_{Q_{0}}>0\quad\text{for}\quad
Q>Q_{0}$ (40)

are all satisfied Brodsky1 ; Brodskya .

(a) Left Panel (b) Right Panel

Figure 5: A graph of $\beta(Q^{2})$ against $Q$ for different $m_{\phi}$
(left) and $m_{\phi}\rightarrow\infty$ (right).

The graphs in the left panel (red, blue, and black) for different $m_{\phi}$
are compared with the results for $\beta(Q)$ for AdS and Modified AdS
presented in Refs. Brodskya ; ads-beta-f  however, they agree better with the
Modified AdS. The transition varies depending on the magnitude of the glueball
mass present. For $m_{\phi}\,=\,1.73\,\text{GeV}$, the transition is at
$Q_{0}\,\approx\,1.41\,\text{GeV}$, for $m_{\phi}\,=\,0.98\text{GeV}$;
$Q_{0}\,\approx\,0.80\,\text{GeV}$ and $m_{\phi}\,=\,1.37\,\text{GeV}$;
$Q_{0}\,\approx\,1.12\,\text{GeV}$. After $Q_{0}$ is reached, the graph starts
decreasing in magnitude with increasing $Q$, representing asymptotic freedom.
The graph in the right panel shows an increase in the transition momentum
$Q_{0}$ for increasing $m_{\phi}\,\rightarrow\,\infty$.

Figure 6: A unified diagram from $\alpha_{s}^{AdS}(Q^{2})$ in both the IR
(non-pQCD) and the UV (pQCD) regions.

A schematic diagram showing the behavior of the strong running coupling in
both the UV (dashed blue) and IR (red) regions in the model framework. This
diagram shows the transition point at $0.89\,\text{GeV}$ (the two graphs 1
(left) and 4(left) intersect at this point), corresponding to
$m_{\phi}\,\approx\,1.09\,\text{GeV}$.

##  VI Analysis and Conclusion

###  VI.1 Analysis

The graphs plotted for the couplings determined from the UV region were for
$f_{\alpha}\,=\,0.50\,\text{GeV}$ and $m_{\phi}\,=\,0.86\,\text{GeV}$. On the
other hand, the couplings determined from the IR region were plotted for
$m_{\phi}\,=\,1.73\,\text{GeV}$. Generally, we considered the lightest scalar
glueball mass consistent with the background of the study presented in Sec.
III. From QCD lattice simulations results, QCD sum rule and QCD phenomenology,
the lightest scalar glueball mass with quantum number $J^{PC}\,=\,0^{++}$ and
frequency $f_{0}(1710)$, has been determined to be
$m_{\phi}\,=\,1.73\,\text{GeV}$ Issifu1 ; Issifu2 ; Issifu3 (and references
therein). Other glueball masses with the same quantum number and frequencies
$f_{0}(500),\,f_{0}(980),\,f_{0}(1370)$ have also been identified Ochs ; Crede
. Despite the negative sign of $m_{\phi}^{2}$ determined in the UV region, its
magnitude is half the one determined in the IR region — see below Eq.(IV.1),
and above Eq.(V) — this accounts for the discrepancies in the magnitude of
$m_{\phi}$ in the UV and the IR regions. Studies in Refs. Kochelev ; Mathieu ;
Miller  shows that the magnitude of the scalar glueball mass determined in the
QGP region is less than the one determined in the IR region. We established
connection between the couplings ($\alpha_{s}^{AdS}(Q^{2})$ and
$\beta(Q^{2})$) and $m_{\phi}$ in both regions. Additionally, we determined
the position of the Landau pole to be
$Q_{\Lambda}\,\approx\,0.58\,\text{GeV}$, Eq.(31), it is also related to the
glueball mass present. Again, we established a relationship between the
transition momentum $Q_{0}$ and $m_{\phi}$, $Q_{0}\,=\,\sqrt{(2/3)}m_{\phi}$,
for the couplings determined in the IR region. For
$m_{\phi}\,=\,1.73\,\text{GeV}$ we determined,
$Q_{0}\,\approx\,1.41\,\text{GeV}$, so glueballs with larger masses are more
likely to be confined than the lighter ones. Although the couplings do not
directly involve quark masses, it is known that; heavy quarks are more likely
to confine than the lighter ones, similar to the behavior of the glueballs
identified here. That implies that relatively heavier glueballs may be
involved in confining heavy quarks and vice versa. We intend to investigate
this phenomenon by fitting a potential model with dependence on glueballs into
the heavy quarkonium spectroscopy to determine the involvement of glueballs in
quark confinement at various masses.

This paper introduces a new perspective to determining the AdS/QCD strong
running coupling and the associated $\beta$-function. Instead of determining
$\alpha_{s}^{AdS}(Q^{2})$ in the less understood IR region by IR distortion of
the AdS space and extrapolating the results to the relatively well-understood
UV region. We introduced a $G(\phi)$ associated with tachyons that distort the
AdS space in two different regions, i.e. when the tachyons are in their free
state and when they are in a condensed state corresponding to UV and IR
deformations of the AdS space respectively. The results obtained, compared to
the well-measured effective charge $\alpha_{s,g1}$ determined from the Bjorken
sum rule for pQCD and AdS/QCD couplings. We established that the coupling
determined from the UV region has a sinusoidal fall off
$\alpha_{s}^{AdS}(Q^{2})\sim{1}/{\sin(\pi Q^{2}/2m_{\phi}^{2})}$, compared to
the pQCD with logarithmic fall off
$\alpha_{s}(Q^{2})\,\sim\,{1}/{\ln(Q^{2}/\Lambda^{2}_{QCD})}$. On the other
hand, the AdS/QCD model with positive dilaton background leads to exponential
fall off $\alpha^{AdS}_{s}(Q^{2})\,\sim\,e^{-Q^{2}/4\kappa^{2}}$; in our case,
the fall-off is $1/(2m^{2}_{\phi}+Q^{2})^{3}$. Indeed, there seems to be an
agreement that the behavior of the QCD running coupling in the IR domain is
not well-defined because different experiments formulated on momentum space
yield different outcomes gp1 ; Deur . Thus, computations of hadronic
characteristics remain the only viable approach for comparing lattice
simulation schemes coupling1 . We established that the $\beta$-functions
determined from the model framework satisfy ”the principle of maximum
conformality” in both UV and IR regions.

The UV deformation of the AdS space studied in Sec. IV shows the appearance of
nonphysical Landau singularity in $\alpha_{s}^{AdS}(Q^{2})$, with the Landau
pole at $Q_{\Lambda}$. On the contrary, the IR deformation, Sec. V, shows a
’regular’ behavior deep in the IR and far UV regions, as expected. The
’regular’ behavior of $\alpha_{s}$ in the IR region is known as the IR
freezing. It is required to fully understand $\alpha_{s}$ since experimental
coupling-x  and lattice results coupling-l  show a usual behavior in both the
UV and IR regions contrary to the theoretical results. Consequently, the
essence of confinement is to make $Q$ finite in all regions, eliminating
Landau ghost pole and IR renormalons confinement-1 . The singularity observed
in $\alpha_{s}$ marks the breakdown of perturbative QCD theory. Hence it is
frequently associated with self-interacting gluons at low energies with hadron
degrees of freedom. In that view, at $Q^{2}\,\rightarrow\,0$ the gluons
dynamically acquire mass $Q^{2}\,=\,-{q}^{2}\,\cong\,m_{A}^{2}$ g-mass ; IR-
fix  which increases $\alpha_{s}$ infinitely. Thus, we can fix the IR freezing
point IR-freezing-ph ; IR-freezing-lt  by introducing $m_{A}^{2}$ into $Q^{2}$
such that its effect is negligible at higher energies
$Q^{2}\,\rightarrow\,\infty$ but has dominant effect at
$Q^{2}\,\rightarrow\,0$ beta-f2 . That will regulate the IR singularity and
enhance the qualitative understanding of color confinement. It is important to
note that; the dynamically generated gluon masses have no association with
spontaneous chiral symmetry breaking. The forces between gluons are;
sufficiently strong enough to generate bound states, especially
$J^{PC}=0^{++}$ color singlet glueballs described by scalar fields. Their
vacuum expectation values are associated with the gluon mass and gluon
condensate $\langle G^{\mu\nu}G_{\mu\nu}\rangle$. Similar to the approach
adopted in this paper. The gluon mass $m_{A}$ have been determined from the
leading order of Yang-Mills theory with auxiliary field $\phi$ to have a
direct relation with the scalar glueball mass $m_{\phi}$ through,
$m_{A}\,=\,m(0^{++})/\sqrt{6}$ Cornwall ; Kondo , where $m(0^{++})$ is the
fluctuations around $\phi$. Similar relations have been reached in Refs.
Issifu2 ; Issifu3  to be $m_{A}\,=\,m_{\phi}/2$, using models with color
dielectric function. In this case, $\alpha_{s}^{AdS}(Q^{2})$ determined from
the UV region of the AdS space takes the form

$\alpha^{AdS}_{s}(Q^{2}+m_{A}^{2})=\dfrac{\sigma\pi}{m_{\phi}^{6}}\left({m_{\phi}^{2}f_{\alpha}^{2}}\right)^{(Q^{2}+m_{A}^{2})/2m_{\phi}^{2}}(2m_{\phi}^{2}-(Q^{2}+m_{A}^{2}))\csc\left(\dfrac{\pi(Q^{2}+m_{A}^{2})}{2m_{\phi}^{2}}\right).$
(41)

For $m_{\phi}\,=\,0.86$ we get $m_{A}\,=\,0.43$ which leads to a finite IR
freezing point as presented in Fig. 7. On the other hand, the IR model self-
regulates at low $Q^{2}$ with a finite freezing point normalized at
$\alpha^{AdS}_{s}(0)\,=\pi$, on $g_{1}$ scheme AdS-book .

Figure 7: A graph of $\alpha_{s}^{AdS}(Q^{2}+m_{A}^{2})/\pi$ (dashed blue) and
$\alpha_{s}^{AdS}(Q^{2})/\pi$ (red) against $Q$ for $m_{A}=0.43\,\text{GeV}$
and $m_{\phi}=0.86\,\text{GeV}$.

The dashed blue graph is well behaved in the IR region due to the IR freezing
point while the red graph shows the singularity in UV region. However, in the
limit of high $Q$ they both fall off in a similar manner.

Figure 8: A graph of $\beta(Q^{2}+m_{A}^{2})$ (dashed blue) and $\beta(Q^{2})$
(red) against $Q$ for $m_{A}=0.43\,\text{GeV}$ and
$m_{\phi}=0.86\,\text{GeV}$.

We compare $\beta$-function with IR freezing to the one without IR freezing.

###  VI.2 Conclusion

We established a connection between tachyonic field $\phi$ and strong
interacting glueball field $\varphi$ in Sec. III and developed; an action for
tachyonic AdS/QCD from DBI action in the subsequent subsections to set the
grounds for the investigation. We determined AdS/QCD running coupling
$\alpha_{s}^{AdS}(Q^{2})$ and the associated $\beta$-function $\beta(Q^{2})$
by distorting the $\text{AdS}_{5}$ space in both UV and IR regions using color
dielectric function. The free tachyon $G(\phi)$ deforms the AdS space in the
UV region, while the condensed tachyon $G(\eta)$ distorts the AdS space in the
IR region. In Sec. IV we determined $\alpha_{s}^{AdS}(Q^{2})$ and
$\beta(Q^{2})$ through the UV distortion of the AdS space and found a good
agreement between the analytical behaviors of our result with both the pQCD
and $\alpha_{s,g1}$ effective charge. The results are presented in Figs. 1, 2
and 3 and compared. We determined that increasing the glueball mass $m_{\phi}$
goes into strengthening color confinement and increases the transitional
momentum. In Sec. V, we introduced condensed tachyon color dielectric function
$G(\eta)$ and used it to deform the AdS space in the IR region. We determined
$\alpha_{s}^{AdS}(Q^{2})$ and $\beta(Q^{2})$ in this region and compared the
results with the one obtained from AdS/QCD through IR distortion of the AdS
space with a positive dilaton profile. The results displayed in Figs. 4 and 5.
The behavior of $\alpha_{s}^{AdS}(Q^{2})$ and $\beta(Q^{2})$; at large
glueball mass limits were studied. We established that glueballs with larger
masses are more likely to confine than lighter ones. A diagram showing a
unified $\alpha_{s}^{AdS}(Q^{2})$ for both IR and UV regions displayed in Fig.
6. This diagram shows a transition at $0.89\,\text{GeV}$ corresponding to a
glueball mass $m_{\phi}\,\approx\,1.09\,\text{GeV}$. We also observed Landau
singularity in $\alpha_{s}^{AdS}(Q^{2})$ in the UV region with its pole at
$Q_{\Lambda}$ and proposed how to fix it. We introduced dynamically generated
gluon mass $m_{A}^{2}$ into $Q^{2}$ to correct the nonphysical behavior at
$Q\,\rightarrow\,0$ limit, and the results presented in Figs. 7 and 8. Other
important results obtained from the model framework are; relationship between
$\alpha_{s}^{AdS}(Q^{2})$, $\beta(Q^{2})$ and $m_{\phi}$ in both regions.
Subsequently, we determined that the transition momentum $Q_{0}$ is directly
related to $m_{\phi}$ as well. We determined that larger glueball masses are
more likely to confine than lighter ones in relatively high momentum transfer
regions. We have also shown that with the appropriate function, particularly,
Higgs-like tachyon potential, one can deform the AdS space in the UV region
with similar characteristics as pQCD. Finally, we intend to extend these
investigations to tachyonic AdS/QCD couplings in both UV and IR regions at a
finite temperature. Again, we intend to study the hadronic spectra and the
hadron form factor using the $G(\eta)$ ansatz since it falls within the IR
region.

###### Acknowledgements.

This work is partly supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) project No.: 168546/2021-3, Brazil.

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\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope}
\end{tikzpicture} \qquad \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (D) {$D$};
\draw[<->, dashed, thick] (B) -- (Y);
\draw[<->, dashed, thick] (D) -- (Y);
\draw[<->, dashed, thick] (B) -- (D);
\end{scope} \end{tikzpicture}}^{\,\color{red} 0} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}\}.\]

* The minimal sufficient adjustment set $\{B,D\}$ is found:

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick, color=red] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, dashed, thick] (D) -- (B);
\draw[<->, dashed, thick] (D) -- (C);
\draw[<->, dashed, thick] (D) -- (X);
\draw[<->, dashed, thick] (D) -- (Y);
\end{tikzpicture}}^{\,\color{red} 3} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, {\color{blue} \{B, D\} } \}.\]

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, dashed, thick] (D) -- (B);
\draw[<->, dashed, thick] (D) -- (C);
\draw[<->, dashed, thick] (D) -- (X);
\draw[<->, dashed, thick, color=red] (D) -- (Y);
\end{tikzpicture}}^{\,\color{red} 2} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B, D\} \}.\]

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, dashed, thick] (D) -- (B);
\draw[<->, dashed, thick, color=red] (D) -- (C);
\draw[<->, dashed, thick] (D) -- (X);
\end{scope} \end{tikzpicture}}^{\,\color{red} 1} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B,D\}\}.\]

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, dashed, color=red, thick] (D) -- (B);
\draw[<->, thick, color=blue] (D) -- (C);
\draw[<->, dashed, thick] (D) -- (X);
\end{scope} \end{tikzpicture}}^{\,\color{red} 2} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B,D\}\}.\]

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (D) -- (C);
\draw[<->, dashed, thick, color=red] (D) -- (X);
\end{scope} \end{tikzpicture}}^{\,\color{red} 1} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B,D\}\}.\]

* Because $D \noconfarc X \mid B, C$, we have

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\node[rv, above of=B, xshift=0.8cm, yshift=-0.3cm] (D) {$D$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (D) -- (C);
\end{scope} \end{tikzpicture}}^{\,\color{red} 0} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \begin{scope}[xshift=3cm]
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B,D\}\}.\]

* Set $\{B,C,D\}$ is identified again as a sufficient adjustment set. This leaves $\mathcal{R}$ unchanged:

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\node[rv, above of=X, yshift=-0.3cm] (B) {$B$};
\node[rv, above of=Y, yshift=-0.3cm] (C) {$C$};
\draw[<->, thick, color=blue] (B) -- (X);
\draw[<->, dashed, thick] (C) -- (X);
\draw[<->, thick, color=blue] (C) -- (Y);
\draw[<->, thick, color=blue] (B) -- (C);
\end{scope} \end{tikzpicture}}^{\,\color{red} \infty} \qquad \begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}
\;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B,D\}\}.\]

\[\mathcal{Q} = \biggl[\; \boxed{\begin{tikzpicture}
\tikzset{rv/.style={circle,inner sep=1pt,draw,font=\sffamily},
node distance=12mm, >=stealth}
\begin{scope}
\node[rv] (X) {$X$};
\node[rv, right of=X, xshift=0.5cm] (Y) {$Y$};
\draw[<->, thick, color=blue] (X) -- (Y);
\end{scope} \end{tikzpicture}}^{\,\color{red} \infty} \;\biggr], \quad \mathcal{R}=\{\{B,C,D \}, \{B, D\} \}.\]

* Upon $\mathcal{Q} = []$, the algorithm terminates and returns

\[\mathcal{R}=\{\{B,C,D \}, \; \{B, D\}\},\]

which contains the only minimal sufficient adjustment set $\{B,D\}$.