# Yang-Mills Casimir wormholes in $D=2+1$

Alana C. L. Santos<EMAIL_ADDRESS>Universidade Federal do Ceará
(UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Célio R. Muniz
<EMAIL_ADDRESS>Universidade Estadual do Ceará (UECE), Faculdade de
Educação, Ciências e Letras de Iguatu, Av. Dário Rabelo s/n, Iguatu-CE,
63.500-00 - Brazil.  Roberto V. Maluf<EMAIL_ADDRESS>Universidade
Federal do Ceará (UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Departamento
de Física Teórica and IFIC, Centro Mixto Universidad de Valencia - CSIC.
Universidad de Valencia, Burjassot-46100, Valencia, Spain.

(August 28, 2024)

###### Abstract

This work presents new three-dimensional traversable wormhole solutions
sourced by the Casimir density and pressures related to the quantum vacuum
fluctuations in Yang-Mills (Y-M) theory. We begin by analyzing the
noninteracting Y-M Casimir wormholes, initially considering an arbitrary state
parameter $\omega$ and determine a simple constant wormhole shape function.
Next, we introduce a new methodology for deforming the state parameter to find
well-behaved redshift functions. The wormhole can be interpreted as a
legitimate Casimir wormhole with an expected average state parameter of
$\omega=2$. Then, we investigate the wormhole curvature properties, energy
conditions, and stability. Furthermore, we discover a novel family of
traversable wormhole solutions sourced by the quantum vacuum fluctuations of
interacting Yang-Mills fields with a more complex shape function. Deforming
the effective state parameter similarly, we obtain well-behaved redshift
functions and traversable wormhole solutions. Finally, we examine the energy
conditions and stability of solutions in the interacting scenario and compare
to the noninteracting case.

Casimir effect. Wormholes. 2+1 dimensions. Yang-Mills.

##  I Introduction

The Casimir effect is associated with quantum vacuum fluctuations when we
impose boundary conditions on quantum fields. The first configuration
analyzed, two neutral parallel conducting plates separated by a very short
distance, produces an attractive force Casimir:1948dh . However, the nature of
this force is strongly dependent on the geometry and dimension in question and
can be repulsive, for example, Boyer:1968uf . This manifestation of quantum
effects on macroscopic scales was confirmed in 1958 Sparnaay:1958wg  and has
been subjected to increasingly precise experiments C1 ; C2 ; C3 . Since then,
this effect has been intensely studied in the literature from different
perspectives Santos:2021jjs ; Muniz:2015jba ; Lima:2019pbo ; Maluf:2019ujv ,
including from generalizations of classical electromagnetism Garattini:2021kca
; deOliveira:2022hew . In this sense, we highlight the analyses performed in
the context of a non-Abelian gauge theory in $2+1$ dimensions via Hamiltonian
formulation Karabali:2018ael , and lattice simulations based on first
principles Chernodub:2018pmt  in which the authors identify vacuum
backreaction effects on the plates.

On the other hand, a distinct characteristic of the Casimir energy density is
the violation of energy conditions due to its negative value for some
configurations. This fact promotes it as one of the scarce energy sources
capable of sustaining traversable wormholes Tripathy:2020ehi  \- solutions of
Einstein’s equations analogous to a hypothetical tunnel in spacetime between
two distant points in the universe, whose traversability requires exotic
matter Morris:1988cz  (except for some modified theories Pavlovic:2014gba ,
which has been useful for detecting these structures DeFalco:2021klh ;
DeFalco:2021ksd ). Interestingly, the pioneering solution in $3+1$ dimensions
was only found in 2019, with the careful work of Garattini Garattini:2019ivd
and, in the sequence, extended to $d$ dimensions Oliveira:2021ypz . However,
Alencar, Bezerra, and Muniz showed that in $2+1$ dimensions, the Casimir
energy density and pressure are not capable of sustaining this structure.
Therefore, we would not have “pure” Casimir wormholes in this reduced
dimension Alencar:2021ejd .

Working with gravity models in lower dimensions makes sense since the
phenomenon of spontaneous dimensional reduction of spacetime has garnered
considerable attention in recent years. It refers to the intriguing notion
that the number of dimensions in our familiar four-dimensional spacetime may
decrease at extremely high energy scales. This concept challenges our
conventional understanding of the fabric of the Universe, suggesting that at
extreme conditions, the fundamental structure of spacetime may undergo a
remarkable transformation Carlip:2016qrb .

In this sense, inspired by works in which quarkonic matter is a source for
wormholes Lobo , we will seek to approach this question again, but considering
a gluonic scenario, through the potentials found for short and long distances
between perfect chromoelectric conductors in the Yang-Mills theory mentioned
before. As we will see, we again find a non-finite redshift function
throughout space, which compromises one of the traversability conditions
established by Morris and Thorne Morris:1988cz . Some methods adopted in the
literature are associated with modifications in the metric Garattini:2019ivd
or in the Casimir radial density/pressure Alencar:2021ejd . In this work, we
propose a new method that makes a slight distortion in the equation of state
by adding a function obeying an inverse power law of the radial coordinate. By
averaging our deformation, we identify that the original equation of state is
maintained, and therefore, we would have a legitimate Casimir source on
average. We will use this approach for noninteracting and interacting
(confined) Yang-Mills fields. It is worth mentioning that Chew and Lim have
recently obtained numerical solutions for wormholes using the phantom field to
support the throat in the Bogomol’nyi-Prasad-Sommerfield limit and beyond, for
the Einstein-Yang-Mills-Higgs action Chew:2021vxh ; Chew:2020svi .

Our paper is organized as follows: In Sec. II, we find a family of traversable
wormhole solutions considering the usual form of Casimir energy density and
our deformation method. Besides, we identified the characteristics of the
matter sources that produce these solutions. In Sec. III, we use our
deformation method again to find new wormhole solutions with Casimir energy
for the $SU(N)$ gauge theory, and we investigate the properties of the
geometry and matter-energy content in these solutions. Finally, we outline our
perspectives and conclusions in Sec. IV.

##  II Noninteracting Case

In this section, we will revisit the issue of whether it is feasible to
maintain a traversable wormhole in a $(2+1)D$ spacetime using the Casimir
quantities - energy density and pressure - as observed in the $(3+1)D$
scenario Garattini:2019ivd . This question was first addressed by Alencar et
al. in Ref. Alencar:2021ejd  in the context of a Casimir wormhole embedding in
a graphene sheet. As we will see below, the answer to this question is
negative, at least if we consider the usual $(2+1)D$ Casimir energy and
pressure. To overcome this limitation, we will present a new method for
generating traversable wormholes in this scenario.

Our starting point is the Einstein field equations in 2+1 dimensions:

$G_{\mu\nu}=\kappa T_{\mu\nu},$ (1)

where $\kappa=8\pi\tilde{G}/c^{4}$ with $\tilde{G}$ being the Newtonian
gravitational constant in two spatial dimensions. In the natural system of
units, $\tilde{G}$ has units of length $[\tilde{G}]=\ell$. For simplicity, we
consider this arbitrary constant to have a magnitude equal to 1. In addition,
$T_{\mu\nu}$ represents the energy-momentum for the matter source.

Let us assume a circularly symmetric and static ansatz for the spacetime
metric with the line element given by

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{1}{1-\frac{b(r)}{r}}dr^{2}+r^{2}d\theta^{2},$
(2)

describing a (2+1)-dimensional Morris-Thorne wormhole in which the redshift
function $\Phi(r)$ and the shape function $b(r)$ are arbitrary functions of
the polar coordinate $r\in\left[r_{0},+\infty\right)$ Perry:1991qq . Thus, the
coordinate $r$ must be decreases from infinity to a minimum value $r_{0}$ ,
the radius of the throat. These functions must satisfy some properties to
ensure a traversable wormhole solution: $(i)$ a flaring-out condition,
determined by the minimality of the wormhole throat, such that
$(b-b^{\prime}r)/b^{2}>0$, and at the throat, $b(r_{0})=r_{0}$; $(ii)$ the
condition $1-b/r\geq 0$ must also satisfy to guarantee wormhole solutions;
$(iii)$ it must be ensured that there are no horizons present, which are
identified as surfaces with $e^{\Phi}\rightarrow 0$, so that $\Phi(r)$ is
finite everywhere Morris:1988cz .

The Einstein field equations under the metric ansatz (2) have the simple
forms:

$\displaystyle G_{\ t}^{t}=$
$\displaystyle\frac{rb^{\prime}-b}{2r^{3}}=\kappa\rho(r),$ (3) $\displaystyle
G_{\ r}^{r}=$ $\displaystyle\frac{\left(r-b\right)\Phi^{\prime}}{r^{2}}=\kappa
p_{r}(r),$ (4) $\displaystyle G_{\ \theta}^{\theta}=$
$\displaystyle\left(1-\frac{b}{r}\right)\left[\Phi^{\prime\prime}+(\Phi^{\prime})^{2}\right]+\frac{\left(b-rb^{\prime}\right)\Phi^{\prime}}{2r^{2}}=\kappa
p_{\theta}(r),$ (5)

where $T_{\ \nu}^{\mu}=\mbox{diag (-$\rho,p_{r},p_{\theta}$)}$ in which
$\rho(r)$ is the surface energy density, $p_{r}(r)$ is the radial pressure,
and $p_{\theta}(r)$ is the lateral pressure. Additionally, the covariant
energy-momentum conservation law leads to

$p_{r}^{{}^{\prime}}=\frac{1}{r}(p_{\theta}-p_{r})-(\rho+p_{r})\Phi^{{}^{\prime}}.$
(6)

From Eq. (3), we note that the flare-out condition $(i)$ is verified if
$\rho(r)<0$. Remembering that the Casimir effect is typically produced by a
negative energy density, we could argue that Casimir energy is a natural
exotic matter source for traversable wormholes. Hence, we will assume that our
matter source is due to the Casimir energy density, following the ideas of
Garattini in Ref. Garattini:2019ivd  closely.

Usually, the Casimir energy density in $(2+1)$-spacetime dimensions can be
represented in the form

$\rho(r)=-\frac{\lambda}{r^{3}},$ (7)

where $\lambda$ is a positive constant which depend on the specific model
considered. For instance, an abelian gauge field confined between two parallel
static wires exhibits a Casimir parameter $\lambda=\zeta(3)/16\pi$, while the
same geometry applied to the non-Abelian $SU(N)$ gauge field leads to
$\lambda=(N^{2}-1)\zeta(3)/16\pi$ (noninteracting case), where $\zeta(x)$ is
the zeta function Chernodub:2018pmt .

Once the Casimir force is obtained by the differentiation of the Casimir
Energy, it is straightforward to identify the Casimir radial pressure as
Alencar:2021ejd

$p_{r}^{\text{Cas}}(r)=-2\frac{\lambda}{r^{3}},$ (8)

which leads to an equation of state (EoS) given by $p_{r}=\omega\rho$ with
$\omega=2$. For our present purpose, let us relax this condition and work with
a generic $\omega$, taking the Casimir limit at an opportune time.

From (3) and (7), we obtain the following shape function

$b(r)=2\kappa\lambda+\left(1-\frac{2\kappa\lambda}{r_{0}}\right)r,$ (9)

where we take into account the boundary condition $b(r_{0})=r_{0}.$
Substituting (9) into (4) and using the EoS $p_{r}=\omega\rho$, we find the
redshift function in the form

$\Phi(r)=\Phi_{0}+\frac{\omega}{2}\left[\ln r-\ln(r-r_{0})\right],$ (10)

where $\Phi_{0}$ is an arbitrary integration constant. In order to obtain the
appropriate asymptotic flat limit, it is convenient to fix $\Phi_{0}=0$ such
that $\lim_{r\rightarrow\infty}\Phi(r)=0.$

Employing the shape function (9) and the redshift function (10), we can write
the line element (2) in the following way:

$ds^{2}=-\left(\frac{r}{r-r_{0}}\right)^{\omega}dt^{2}+\frac{1}{\frac{2\lambda\kappa}{r_{0}}\left(1-\frac{r_{0}}{r}\right)}dr^{2}+r^{2}d\theta^{2}.$
(11)

Unfortunately, due to the presence of the term $\ln(r-r_{0})$ in (10), the
line element (11) is not expected to represent a traversable wormhole. In
fact, for $\omega>0$, there is no event horizon since
$\left(\frac{r}{r-r_{0}}\right)^{\omega}=0$ does not have a simple zero at
$r=r_{H}$. Therefore, we have neither a black hole nor a traversable wormhole,
as $\Phi(r)$ is not finite everywhere and diverges at $r=r_{0}$. Instead, we
have a singularity described by the following line element (see
Garattini:2019ivd , Appendix A):

$ds^{2}=-\frac{1}{\left(1-\frac{r_{0}}{r}\right)^{\omega}}dt^{2}+\frac{1}{\frac{2\lambda\kappa}{r_{0}}\left(1-\frac{r_{0}}{r}\right)}dr^{2}+r^{2}d\theta^{2}.$
(12)

Furthermore, in the case $\omega<0$, we have one horizon at $r=r_{0}$
($g_{tt}=0$ has a simple zero at $r=r_{0}$), and for the particular value
$\omega=-1$ , we find one black Hole solution given by

$ds^{2}=-\left(1-\frac{r_{0}}{r}\right)dt^{2}+\frac{1}{\frac{2\lambda\kappa}{r_{0}}\left(1-\frac{r_{0}}{r}\right)}dr^{2}+r^{2}d\theta^{2}.$
(13)

As pointed out in Ref. Garattini:2019ivd , the presence of a logarithmic term
in the redshift function is also observed in the $(3+1)$-dimensional case.
However, unlike in that case, we cannot eliminate this “annoying” term because
in $(2+1)$ dimensions, we have less freedom to fix the free parameters of the
system. The metric (11) was also obtained from Ref. Alencar:2021ejd  with
$\omega=-2$, in which the authors added a cosmological constant to work around
the problem that was pointed out. Here, we propose a different strategy. Let
us assume a distortion of the EoS in the following way:

$\omega\rightarrow\omega+\epsilon f(r),$ (14)

where $\epsilon$ is a free parameter that is responsible for deviating
$\omega$ from the original Casimir prescription. The function $f(r)$ can be
chosen to obtain the desired wormhole solution.

With prescription (14), equation (4) assume the following form

$\Phi^{\prime}(r)=-\frac{r_{0}(\omega+\epsilon f(r))}{2r(r-r_{0})}.$ (15)

A simple and convenient choice for the function $f(r)$ able to accomplish our
aforementioned objectives is $f(r)=-1/r$. Therefore, the redshift function
becomes

$\Phi_{\epsilon}(r)=\Phi_{0}+\frac{1}{2r_{0}}\left[\frac{\epsilon
r_{0}}{r}-(\epsilon-\omega r_{0})\ln(r)+(\epsilon-\omega
r_{0})\ln(r-r_{0})\right],$ (16)

and, similar to the $(3+1)$-dimensional case Garattini:2019ivd , the term
$\ln(r-r_{0})$ can be removed by setting the $\epsilon$-parameter as
$\epsilon=\omega r_{0}$. Then, the line element (2) takes the form

$ds^{2}=-e^{\frac{\omega
r_{0}}{r}}dt^{2}+\frac{1}{1-\frac{r_{0}}{r}}dr^{2}+r^{2}d\theta^{2},$ (17)

where $\Phi_{0}=0$, and we fix $r_{0}=2\lambda\kappa=16\pi\lambda\tilde{G}$
for simplicity since the $b(r)$ function assumes a constant form. Now, our
solution (17) satisfies the conditions $(i)-(iii)$ and represents a
traversable wormhole with a throat radius on the order of $r_{0}\sim\tilde{G}$
(in natural units).

It is noteworthy that the above result can be generalized to give rise to a
class of Casimir-like traversable wormholes. This is achieved by taking
$f(r)=-1/r^{n}$, where $n$ is a positive integer. Solving Eq. (15) for
different values of $n$, we find a family of redshift functions
$\Phi_{\epsilon}^{(n)}(r)$ expressed as

$\Phi_{\epsilon}^{(2)}(r)=\Phi_{0}+\frac{1}{4r_{0}^{2}}\left[\frac{\epsilon
r_{0}}{r^{2}}(2r+r_{0})-2(\epsilon-\omega r_{0}^{2})\ln(r)+2(\epsilon-\omega
r_{0}^{2})\ln(r-r_{0})\right],$ (18)
$\Phi_{\epsilon}^{(3)}(r)=\Phi_{0}+\frac{1}{12r_{0}^{3}}\left[\frac{\epsilon
r_{0}}{r^{3}}(6r^{2}+3rr_{0}+2r_{0}^{2})-6(\epsilon-\omega
r_{0}^{3})\ln(r)+6(\epsilon-\omega r_{0}^{3})\ln(r-r_{0})\right],$ (19)

or in the general form

$\displaystyle\Phi_{\epsilon}^{(n)}(r)=$
$\displaystyle\Phi_{0}+\frac{1}{2a_{1}r_{0}^{n}}\left[\frac{\epsilon
r_{0}}{r^{n}}\sum_{p=1}^{n}a_{p}r^{n-p}r_{0}^{p-1}-a_{1}(\epsilon-\omega
r_{0}^{n})\ln(r)+a_{1}(\epsilon-\omega r_{0}^{n})\ln(r-r_{0})\right],$ (20)

where the coefficients $a_{p}(n)$ are determined to satisfy the associated
differential equation (15). Note that the $\ln(r-r_{0})$ term is removed in
each case by setting $\epsilon=\omega r_{0}^{n}$, resulting in the general
wormhole solution given by

$ds^{2}=-\exp\left(\frac{\omega
r_{0}}{a_{1}r^{n}}\sum_{p=1}^{n}a_{p}r^{n-p}r_{0}^{p-1}\right)dt^{2}+\frac{1}{1-\frac{r_{0}}{r}}dr^{2}+r^{2}d\theta^{2},$
(21)

with $n$ being a positive integer. Evidently, our distortion procedure of the
EoS (8) only affects the redshift function, keeping the shape function
solution (9) unaltered.

It is worth commenting that the promotion of the EoS parameter to be a
function of the radial coordinate is a common approach in cosmology in the
attempt to model dark energy Sharov:2022hxi ; Barboza:2009ks . Also, this
approach usually happens in the context of a Brans-Dicke type scalar-tensor
theories, where the EoS parameter $\omega$ depends on the scalar field profile
and hence on the radial coordinate Fujii:2003pa ; Singh:1987is ; Lee:2010tm .
Notably, this framework also led to the discovery of wormhole solutions
Franciolini:2018aad ; Agnese:1995kd ; Lobo:2010sb  . However, our main goal
with our study is to find a simple method to create traversable wormholes in
$D=2+1$ using the Casimir energy as source matter. To this end, we have
proposed to deform the EoS parameter $\omega$ to encompass curvature effects
arising from the backreaction between gravity and the matter source to achieve
a physically acceptable wormhole solution. Note that asymptotically, the
effective EoS parameter follows the usual Casimir relation (8), while for
short distances, it undergoes a correction that enables the formation of
traversable wormholes. In this way, the issue of the physical origins of our
effective EoS parameter is of interest but lies beyond our present scope.

To finish our analysis of the Casimir wormhole solution associated to the line
element (21), let us exhibit the expression for the Kretschmann scalar
$K=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ for $n=1$ and $\omega=2$,
namely,

$K=\frac{r_{0}^{2}}{r^{10}}\left(4r_{0}^{4}+12r_{0}^{3}r-3r_{0}^{2}r^{2}-32r_{0}r^{3}+21r^{4}\right).$
(22)

As expected, the Kretschmann scalar does not diverge on the interval
$r\in\left[r_{0},+\infty\right)$, and approaches zero at the asymptotic limit.
Furthermore, on the throat, it assumes the constant value of
$\left.K\right|_{r=r_{0}}=\frac{2}{r_{0}^{4}}_{.}$. In Fig. 1, we plot the
Kretschmann scalar as a function of the radial coordinate. As expected, when
we turn off our correction of the EoS parameter ($\epsilon=0$), the
Kretschmann scalar diverges at the throat, yielding a naked singularity
described by Eq. (12) with $\omega=2$. On the other hand, when $\epsilon\neq
0$, it remains finite across all considered $n$ values. Additionally, we note
that for $n>1$, the scalar $K$ has one minimum and maximum locally near the
throat. The value of the Kretschmann scalar is a global maximum at the
wormhole throat, increasing with the value of the power, $n$, while there is
also a local maximum that increases with $n$, indicating that the stability of
the wormhole solution decreases for large $n$. The local minimum shifts
slightly with $n$ towards the throat but remains on the $n=1$ curve. Next, we
will analyze the properties of the matter content associated with our wormhole
solution (21).

Figure 1: Kretschmann scalar associated with the metrics (12) ($\epsilon=0$)
and (21) for some values of $n$, with $\omega=2$ and $r_{0}=1$.

###  II.1 Source properties and stability analysis

Now we are interested in studying the properties of the matter content
responsible for forming our family of wormhole solutions (21).

We have observed that our distortion method does not affect the Casimir energy
density $\rho(r)$, which remains with the same expression as in Eq. (7).
However, the original Casimir radial pressure (8) changes to the following
form

$p_{r}^{(n)}(r)=\omega_{r}^{(n)}(r)\rho(r)=\omega\left[1-\left(\frac{r_{0}}{r}\right)^{n}\right]\rho(r).$
(23)

It is worth highlighting the connection between the original EoS and the
modified equation that we have used here. Taking the mean value of
$\omega_{r}^{(n)}$, we obtain

$\left\langle\omega_{r}^{(n)}\right\rangle=\lim_{r^{\prime}\rightarrow\infty}\frac{\omega}{r^{\prime}-r_{0}}\int_{r_{0}}^{r^{\prime}}\left[1-\left(\frac{r_{0}}{r}\right)^{n}\right]dr=\omega,$
(24)

which means that our approach recovers the standard relation between the
density energy and the radial pressure from the Casimir effect, as expressed
by their mean values. Specifically, when $\omega=2$, we obtain the Casimir
relation (8):

$\left\langle
p_{r}^{(n)}\right\rangle=\left\langle\omega_{r}^{(n)}\right\rangle\rho=p_{r}^{\text{Cas}}.$
(25)

Another essential piece of information about the matter content comes from the
lateral pressure $p_{\theta}$. Using the last Einstein equation (5) and
substituting our solutions (9) and (20), we can write (with $\epsilon=\omega
r_{0}^{n}$ and $r_{0}=2\lambda\kappa$):

$p_{\theta}^{(n)}(r)=\omega_{\theta}^{(n)}(r)\rho(r),$ (26)

where $\omega_{\theta}^{(n)}(r)$ has the following explicit form for some
$n$-values:

$\displaystyle\omega_{\theta}^{(1)}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}r_{0}(r_{0}-r)+\omega r(5r_{0}-4r)}{2r^{2}},$
(27) $\displaystyle\omega_{\theta}^{(2)}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}r_{0}(r_{0}-r)(r_{0}+r)^{2}+\omega
r^{2}(7r_{0}^{2}-r_{0}r-4r^{2})}{2r^{4}},$ (28)
$\displaystyle\omega_{\theta}^{(3)}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}r_{0}(r_{0}-r)(r_{0}^{2}+r_{0}r+r^{2})^{2}+\omega
r^{3}(9r_{0}^{3}-r_{0}^{2}r-r_{0}r^{2}-4r^{3})}{2r^{6}}.$ (29)

Note that for any finite value of $n$, the associated
$\omega_{\theta}^{(n)}(r)$ is finite on the throat and vanishes when
$r\rightarrow\infty$. Moreover, we can explicitly verify that the EoM (3-5)
and the conservation law (6) are satisfied by the above expressions.

Figure 2 provides a comprehensive visualization of the energy density, radial
and lateral pressures, and their various combinations to aid in the analysis
of energy conditions. Specifically, the null energy condition (NEC), which
states that the energy density plus any of the pressures cannot be negative
($\rho+p_{i}\geq 0$), and the dominant energy condition (DEC), which states
that the energy density must be greater than or equal to the absolute value of
any of the pressures ($\rho-|p_{i}|\geq 0$), are examined. It is worth noting
that these conditions are not satisfied throughout the domain, as evidenced by
the violation of the NEC and DEC. Furthermore, the strong energy condition
(SEC), which requires the NEC to hold true and $\rho+\sum_{i}p_{i}\geq 0$, is
also not satisfied since NEC is violated.

---
(a) (b)
(c) (d)
Figure 2:  Density and radial pressures (a), density and lateral pressures
(b), and combination of density and pressures (c) and (d) in order to verify
energy conditions, namely, DEC, considering $n=1,2,3$, $\omega=2$, and
$r_{0}=\lambda=1$, in Planck units.

####  II.1.1 Stability from sound velocity

In order to evaluate the stability of the Casimir wormhole family given by Eq.
(21), we must firstly analyze the condition described by the expression
Capozziello:2020zbx ; Capozziello:2022zoz :

$v_{s}^{2}(r)=\frac{1}{2}\left[\frac{d(p_{r}+p_{\theta})}{d\rho}\right]=\frac{1}{2}\left[\frac{p_{r}^{\prime}(r)+p_{\theta}^{\prime}(r)}{\rho^{\prime}(r)}\right]\geq
0,$ (30)

with $v_{s}$ representing the sound velocity in the medium, where we take the
average pressure between the lateral and radial ones in 2+1 dimensions, which
are given by Eqs. (23) and (26). As previously demonstrated, the effective
state parameter associated with the wormhole source is expressed by the
function $\omega^{(n)}(r)=2[1-(r_{0}/r)^{n}]$, which must be shifted from
value $\omega=2$ to enable the traversability of the wormhole. Thus, for
$n=1$, the sound velocity takes the form

$v_{s}^{2(1)}(r)=\omega^{2}\left(\frac{5r_{0}^{2}}{12r^{2}}-\frac{r_{0}}{3r}\right)+\omega\left(\frac{r_{0}}{r}-\frac{1}{2}\right),$
(31)

such that it is independent of $\lambda$, and its value at the throat does not
change with different choices for $r_{0}$. The same behavior is observed for
all $n$ integer finite. The stability of the solutions is predominantly
assured near the wormhole throat, as depicted in Figure 3.

Figure 3: Squared sound velocity as a function of the radial coordinate, for
the noninteracting case, considering $\omega=2$ and $r_{0}=1$.

Although stability is guaranteed, it is essential to acknowledge that the
system is considered unphysical due to the sound speed consistently surpassing
the speed of light near the throat, regardless of the parameter values.
However, it is noteworthy that a superluminal velocity may not necessarily be
forbidden in this context, given the presence of exotic matter responsible for
the wormhole’s formation Ellis:2007ic . Nevertheless, we can address this
concern in the interacting case by carefully selecting appropriate parameter
values.

####  II.1.2 Stability from TOV equation

The stability of compact objects, such as stellar structures, can be also
examined through the Tolman-Oppenheimer-Volkov (TOV) equation, initially
introduced in the context of neutron stars Tolman:1939jz ; Oppenheimer:1939ne
. Hence, the stability conditions for wormholes could be investigated by
employing an equilibrium condition obtained from the TOV equation
Gorini:2008zj ; Kuhfittig:2020fue ; Mustafa:2021bfs ; Sokoliuk:2021rtv .
Combining the Morris-Thorne metric ansatz (2) with the conservation law of
stress-energy tensor $\nabla_{\mu}T^{\mu\nu}=0$, we obtain the following TOV
equation in a $2+1$ dimensional spacetime:

$\frac{dp_{r}}{dr}+\Phi^{\prime}(\rho+p_{r})+\frac{1}{r}(p_{r}-p_{\theta})=0.$
(32)

The equilibrium state of the wormhole structure is determined by three terms
in this equation, namely, the hydrostatic force ($\mathcal{F}_{h}$), the
gravitational force ($\mathcal{F}_{g}$), and the anisotropic force
($\mathcal{F}_{a}$), defined as follows Kuhfittig:2020fue :

$\displaystyle\mathcal{F}_{h}=-\frac{dp_{r}}{dr},\ \ \ \ \
\mathcal{F}_{g}=-\Phi^{\prime}(\rho+p_{r}),\ \ \ \ \
\mathcal{F}_{a}=\frac{1}{r}(p_{\theta}-p_{r}).$ (33)

Thus, Eq. (32) takes the simple form given by:

$\mathcal{F}_{h}+\mathcal{F}_{g}+\mathcal{F}_{a}=0.$ (34)

It is straightforward to verify that our class of traversable wormhole
solutions satisfies the TOV equation for all finite values of $n$. For
instance, when $n=1$, and employing Eqs (7), (17), (23), and (26), we obtain
the following expressions for the forces:

$\displaystyle\mathcal{F}_{h}$ $\displaystyle=$
$\displaystyle-\frac{\lambda\omega(3r-4r_{0})}{r^{5}},$ (35)
$\displaystyle\mathcal{F}_{g}$ $\displaystyle=$
$\displaystyle-\frac{\lambda\omega(r_{0}r+\omega r_{0}r-\omega
r_{0}^{2})}{2r^{6}},$ (36) $\displaystyle\mathcal{F}_{a}$ $\displaystyle=$
$\displaystyle\frac{\lambda\omega(6r^{2}-7r_{0}r++\omega r_{0}r-\omega
r_{0}^{2})}{2r^{6}},$ (37)

such that the equilibrium condition (34) is satisfied for all $r$, and
confirming it as a stable wormhole.

The profiles of $\mathcal{F}_{h}$, $\mathcal{F}_{g}$, and $\mathcal{F}_{a}$
for our wormhole solutions with $n=1$ and $n=2$ are depicted in Fig. 4(a) and
4(b), respectively, with the assigned values of the free parameters as
$\omega=2$ and $\lambda=r_{0}=1$. We can observe that the hydrostatic force
dominates over the gravitational and anisotropic forces. For both cases,
$\mathcal{F}_{h}$ takes positive values near the throat, while
$\mathcal{F}_{g}$ and $\mathcal{F}_{a}$ are negative, clearly indicating that
to maintain the system in an equilibrium state, the hydrostatic force is
balanced by the combined effect of gravitational and the anisotropic forces.
Furthermore, the magnitude of the forces near the throat increases as $n$
grows. This can be interpreted as an effect of increasing curvature with the
value of $n$, as verified earlier in our analysis of the Kretschmann scalar.

---
(a) (b)
Figure 4:  The graphical representation of $\mathcal{F}_{h}$,
$\mathcal{F}_{g}$, and $\mathcal{F}_{a}$ as functions of the radial coordinate
$r$, with $\omega=2$ and $\lambda=r_{0}=1$, is shown in (a) for $n=1$ and in
(b) for $n=2$.

##  III Interacting (confined) case

The Casimir energy of two parallel wires in the $(2+1)$-dimensional Yang-Mills
theory was analyzed using both a Hamiltonian approach Karabali:2018ael  and
lattice simulation Chernodub:2018pmt . Despite the distinct analytical
expressions for the renormalized zero-point energy found in the cited works,
they share common features: Both results show graphs very close to the Casimir
energy profile, and a massive quantity exponentially suppresses the Casimir
energy. This emergent mass scale may be related to the glueball masses and
influences the confinement-deconfinement transition of the gluonic fields
between the wires Pasechnik:2016wkt .

Thus, in the context of our present work, the Casimir energy’s infrared
damping may affect the wormhole solutions behavior, resulting in new physical
effects. These implications have not yet been evaluated in the literature and
will be the main focus of this section.

Effectively, the lattice calculation of the Casimir energy for the $SU(N)$
gauge theory was fitted by the following expression Chernodub:2018pmt

$\frac{\mathcal{E}}{L}=-\mbox{dim}G\frac{\zeta(3)}{16\pi}\frac{\left(R\sqrt{\sigma}\right)^{-\nu}}{R^{2}}e^{-M_{\text{Cas}}R},$
(38)

where $L$ is the length of the wire, $R$ is the distance between them, and
$\sigma$ is the tension of the confining (fundamental) Yang-Mills string at
zero temperature. The exponent $\nu$ represents an anomalous dimension of the
Casimir potential at short distances, and the quantity $M_{\text{Cas}}$
represents the effective Casimir mass associated with the nonperturbative mass
gap at large distances. The free parameters $\nu$ and $M_{\text{Cas}}$ can be
tuned to obtain the best lattice fit. In the case of $SU(2)$, where
$\mbox{dim}G=3$, the best fit for lattice simulations was obtained for
$\nu=0.05(2)$ and $M_{\text{Cas}}=1.38(3)\sqrt{\sigma}$ . As noted by the
authors in Chernodub:2018pmt , the value of $M_{\text{Cas}}$ is considerably
lower than the minimum value estimated for the glueball mass, which is around
$4.7\sqrt{\sigma}$ Karabali:2018ael ; Teper:1998te ; Athenodorou:2016ebg .

Adopting the procedure of the previous sections, let us promote the distance
between the wires to the radial coordinate $r$ so that the Casimir energy
density $\rho(r)$ and the associated radial pressure $p_{r}(r)$ can be
calculated as follows:

$\rho(r)=\frac{\mathcal{E}}{Lr}=-\lambda\frac{\left(r\sqrt{\sigma}\right)^{-\nu}}{r^{3}}e^{-M_{\text{Cas}}r},$
(39)

with $\lambda=\mbox{dim}G\zeta(3)/16\pi$, and

$\displaystyle p_{r}(r)$
$\displaystyle=\frac{F}{L}=-\frac{1}{L}\frac{d\mathcal{E}}{dr},$
$\displaystyle=-\lambda\frac{(2+\nu+M_{\text{Cas}}r)\left(r\sqrt{\sigma}\right)^{-\nu}}{r^{3}}e^{-M_{\text{Cas}}r},$
(40)

where we write $p_{r}(r)=\omega(r)\rho(r)$ with
$\omega(r)=2+\nu+M_{\text{Cas}}r.$ It is worth noting that when the
interactions are turned off, $M_{\text{Cas}}=0$ and $\nu=0$, and the
expressions (39)-(40) reduce to their tree-level form (7)-(8), respectively.

Substituting (39) into (3), we obtain the shape function of the Casimir-Yang-
Mills wormhole as

$b(r)=r+2\kappa\lambda\sigma^{-\frac{\nu}{2}}M_{\text{Cas}}^{\nu+1}r\left[\Gamma(-\nu-1,M_{\text{Cas}}r)-\Gamma(-\nu-1,M_{\text{Cas}}r_{0})\right],$
(41)

where $\Gamma(a,z)$ is the incomplete gamma function
$\Gamma(a,z)=\int_{z}^{\infty}dtt^{a-1}e^{-t}$. Naturally, the constant of
integration was fixed such that the throat condition $b(r_{0})=r_{0}$ is
satisfied.

Figure 5: Influence of the parameters $\nu$ and $M_{\text{Cas}}$ on the
inequality (45). In this graphic, we adopted $r=4$ and $r_{0}=2$ (in natural
units $\hbar=c=\tilde{G}=1$).

Our goal now is to investigate how the Casimir shape function is affected by
non-abelian corrections, which are characterized by the parameters $\nu$ and
$M_{\text{Cas}}$. First, we may note that in the $\nu\rightarrow 0$ and
$M_{\text{Cas}}\rightarrow 0$ limits, the expression (41) reduces to

$\lim_{\begin{smallmatrix}\nu\to 0\\\ M_{\text{Cas}}\rightarrow
0\end{smallmatrix}}b(r)=2\kappa\lambda+\left(1-\frac{2\kappa\lambda}{r_{0}}\right)r,$
(42)

agreeing with our original result (9). Moreover, the flare-out condition $(i)$
is satisfied for all range of parameters, namely,

$\frac{b-rb^{\prime}}{b^{2}}=\frac{2\kappa\lambda\sigma^{\nu/2}r^{-\nu-2}e^{-M_{\text{Cas}}r}}{\left[2\kappa\lambda
M_{\text{Cas}}^{\nu+1}\left(\Gamma(-\nu-1,M_{\text{Cas}}r)-\Gamma(-\nu-1,M_{\text{Cas}}r_{0})\right)+\sigma^{\nu/2}\right]^{2}}>0,$
(43)

and on the throat

$b^{\prime}(r_{0})=1-2\kappa\lambda\sigma^{-\frac{\nu}{2}}r_{0}^{-\nu-1}e^{-M_{\text{Cas}}r_{0}}<1,$
(44)

since $\kappa$, $\lambda$, $\sigma$, $\nu$, and $M_{\text{Cas}}$ are positive
real quantities. Also, the property $(ii)$ $1-b/r\geq 0$ holds true if the
inequality

$\Delta\Gamma\equiv\Gamma(-\nu-1,M_{\text{Cas}}r)-\Gamma(-\nu-1,M_{\text{Cas}}r_{0})\leq
0,$ (45)

is satisfied, which is always the case for $r\geq r_{0}$. The influence of the
$\nu$ and $M_{\text{Cas}}$ parameters on the inequality (45) is indicated in
Fig. 5, where we take $r=4$ and $r_{0}=2$. As we can see, for large values of
$M_{\text{Cas}}$, the difference in (45) goes to zero.

The asymptotic flatness condition, $\lim_{r\rightarrow\infty}(1-b(r)/r)=1$,
implies that

$1-2\kappa\lambda\sigma^{-\frac{\nu}{2}}M_{\text{Cas}}^{\nu+1}\Gamma(-\nu-1,M_{\text{Cas}}r_{0})=0,$
(46)

which involves solving a transcendental equation for $r_{0}$. This equation
does not have an analytic solution for arbitrary values of $\nu$, $\sigma$,
and $M_{\text{Cas}}$. To get some insight into the effect of Yang-Mills
corrections on the throat radius $r_{0}$, we can expand the equation (46) for
small values of $\nu$ and $M_{\text{Cas}}$. The result can be written as

$r_{0}-2\kappa\lambda\left\\{1-M_{\text{Cas}}r_{0}\left[1-\gamma_{\text{E}}-\ln\left(M_{\text{Cas}}r_{0}\right)\right]-\frac{\nu}{2}\left[2+\ln\left(\sigma
r_{0}^{2}\right)\right]\right\\}=0,$ (47)

where $\gamma_{\text{E}}$ is the Euler-Mascheroni constant, with numerical
value $\gamma_{\text{E}}\approx 0.577216$. Notice that we recover our tree-
level result $r_{0}^{\text{\mbox{tree}}}=2\kappa\lambda$ for $\nu=0$ and
$M_{\text{Cas}}=0$. Furthermore, the non-abelian interaction causes a decrease
in the throat radius, which may eventually result in its closure.

Figure 6: Numerical solution for (46) with respect to the parameters $\nu$ and
$M_{\text{Cas}}$. In this graphic, we adopted $\sqrt{\sigma}=1$,
$\lambda=3\zeta(3)/16\pi$ and $\kappa=8\pi$ (in natural units
$\hbar=c=\tilde{G}=1$). Figure 7: Numerical solution for (46) with respect to
the parameter $\sqrt{\sigma}$. In this plot, we use $\lambda=3\zeta(3)/16\pi$,
$\kappa=8\pi$ and fix $\nu=0.05$ and $M_{\text{Cas}}=1.38\sqrt{\sigma}$ (in
natural units $\hbar=c=\tilde{G}=1$).
---
(a) (b)
Figure 8: Embedded shapes of the Casimir-Yang-Mills wormhole: with
$M_{\text{Cas}}=0.001$ and $r_{0}=3.1841$ (a), with $M_{\text{Cas}}=0.1$ and
$r_{0}=1.9602$ (b). In both cases we fix $\nu=0.05$ and $\sqrt{\sigma}=1$.
Also, we assumed $\lambda=3\zeta(3)/16\pi$ and $\kappa=8\pi$ (in natural units
$\hbar=c=\tilde{G}=1$).

In order to determine the dependence of the throat radius $r_{0}$ on the Yang-
Mills parameters, namely, the string tension $\sigma$, anomalous dimension
$\nu$, and the Casimir mass $M_{\text{Cas}}$, we consider two separate cases.
First, we fix $\sqrt{\sigma}=1$ and consider the running of $\nu$ and
$M_{\text{Cas}}$. The resultant dependence of $r_{0}$ with respect to $\nu$
and $M_{\text{Cas}}$ is depicted in Fig. 6, where we adopt
$\lambda=3\zeta(3)/16\pi$ and $\kappa=8\pi$. As we can see, when the
parameters increase, the radius $r_{0}$ decreases, as expected from our
perturbative result (47). Next, we assume the best-fit values of the
parameters $\nu$ and $M_{\text{Cas}}$ for the $SU(2)$ gauge group as
calculated via lattice techniques in Ref. Chernodub:2018pmt . Specifically, we
set $\nu=0.05$ and $M_{\text{Cas}}=1.38\sqrt{\sigma}$, and introduce the
variable $x_{0}=r_{0}\sqrt{\sigma}$ to numerically solve Eq. (46). The
numerical results are presented in Fig. 7. The graph shows that as
$\sqrt{\sigma}$ increases, the value of $x_{0}$ increases gradually. This
outcome can be attributed to the stretching of the wormhole throat, induced by
the progressive increase in string tension, which is designed to release the
gluons from their confined state. The dependence of the throat radius on the
Casimir mass is also visualized by examining the embedding geometry, as
illustrated in Figure 8.

Let us turn our attention to the redshift function associated with the
Casimir-Yang-Mills wormhole. By replacing the radial pressure (40) and the
shape function (41) into Eq. (4), we find

$\Phi(r)=\Phi_{0}+\int_{r_{0}}^{r}\frac{\left(2+\nu+M_{\text{Cas}}\bar{r}\right)\left(M_{\text{Cas}}\bar{r}\right){}^{-\nu-1}e^{-M_{\text{Cas}}\bar{r}}}{2\bar{r}\left[\Gamma\left(-\nu-1,M_{\text{Cas}}\bar{r}\right)-\Gamma\left(-\nu-1,M_{\text{Cas}}r_{0}\right)\right]}d\bar{r},$
(48)

in which the integration is not possible to solve algebraically. Therefore, we
performed numerical integration for different values of throat radius with
$x=r\sqrt{\sigma}$, as shown in Fig. 9. As we can see, the redshift function
is divergent on the throat spoiling the traversable condition $(iii)$, a
situation analogous to the previous case study in section II.

Figure 9: Numerical solution for Eq. (48) for some values of
$x_{0}=r_{0}\sqrt{\sigma}$. We assumed $\nu=0.05$,
$M_{\text{Cas}}=1.38\sqrt{\sigma}$, and $\Phi_{0}=0$.

In order to find a traversable wormhole solution, we can apply our deformation
method to Eq. (48) in a similar way to the earlier case. We replaced the EoS
parameter $\omega(r)$ in (40) by

$\omega(r)\rightarrow\omega(r)-\epsilon\frac{M_{\text{Cas}}}{r^{n}},$ (49)

and expanded the integrand in (48) as a power series of $\nu$ and
$M_{\text{Cas}}$, which resulted in

$\Phi_{\epsilon}^{(n)}(r)=\Phi_{0}+\int_{r_{0}}^{r}\frac{r_{0}}{2(\bar{r}-r_{0})^{2}}\left[\frac{(\bar{r}-r_{0})}{\bar{r}^{n+1}}\left(\epsilon
M_{\text{Cas}}-\bar{r}^{n}(2+3\nu-M_{\text{Cas}}\bar{r})\right)+2(\nu-
M_{\text{Cas}}r_{0})\ln\left(\frac{\bar{r}}{r_{0}}\right)\right]d\bar{r},$
(50)

where the higher-order terms have been neglected. The integral can now be
computed for any choice of the positive integer $n$. For example, if we take
$n=1$, we obtain the following $\epsilon$-deformed redshift function:

$\displaystyle\Phi_{\epsilon}^{(1)}(r)$
$\displaystyle=\Phi_{0}+\frac{1}{2r_{0}}\left[\frac{\epsilon
M_{\text{Cas}}r_{0}}{r}+\frac{2r_{0}^{2}\left(M_{\text{Cas}}r_{0}-\nu\right)}{r-r_{0}}\ln\left(\frac{r}{r_{0}}\right)\right.$
$\displaystyle\ \ +\left(r_{0}\left(2+\nu+2M_{\text{Cas}}r_{0}\right)-\epsilon
M_{\text{Cas}}\right)\ln r$ $\displaystyle\left.\
-\left(r_{0}\left(2+\nu+M_{\text{Cas}}r_{0}\right)-\epsilon
M_{\text{Cas}}\right)\ln\left(r-r_{0}\right)\right].$ (51)

Note that the logarithmic coefficients in lines two and three of Eq. (51)
differ due to the Casimir mass correction. Nevertheless, we can eliminate the
term $\ln(r-r_{0})$ that diverges at the throat by fixing the parameter
$\epsilon$ as

$\epsilon=\frac{r_{0}}{M_{\text{Cas}}}\left(2+\nu+M_{\text{Cas}}r_{0}\right).$
(52)

Hence, at the leading order, the linearized redshift function takes the form

$\Phi^{(1)}(r)=\frac{r_{0}}{2r}\left(2+\nu+M_{\text{Cas}}r_{0}\right)+\frac{1}{2}M_{\text{Cas}}r_{0}\ln
r-\frac{r_{0}\left(\nu-
M_{\text{Cas}}r_{0}\right)}{r-r_{0}}\ln\left(\frac{r}{r_{0}}\right),$ (53)

where we set $\Phi_{0}=0$. In proximity of the throat, we find

$\lim_{r\rightarrow
r_{0}}\Phi^{(1)}(r)=1-\frac{\nu}{2}+M_{\text{Cas}}r_{0}\left(\frac{3}{2}+\frac{1}{2}\ln
r_{0}\right),$ (54)

is a finite quantity. Therefore, we have demonstrated that the Casimir-Yang-
Mills energy indeed generates a traversable wormhole in $(2+1)$-dimensions.
Also, we may note that, in the massless limit, the expression (53) agrees with
our tree-level result, namely,

$\lim_{\begin{smallmatrix}\nu\to 0\\\ M_{\text{Cas}}\rightarrow
0\end{smallmatrix}}\Phi^{(1)}=\frac{r_{0}}{r}=\Phi_{\text{tree}}^{(1)}.$ (55)

It is worth noting that in the asymptotic limit as $r\rightarrow\infty$, one
finds that $\Phi^{(1)}(r)$ diverges due to the presence of the term
$\frac{1}{2}r_{0}M_{\text{Cas}}\ln r$ in (53). This is a consequence of the
absence of the factor $\exp(-M_{\text{Cas}}\bar{r})$ in the linearized
integration (50). However, by applying the substitution (49) into Eq. (48) and
fixing the $\epsilon$-parameter as specified in Eq. (52), we can numerically
integrate the resulting equation and observe that the redshift function is
finite at the throat radius and vanishes as $r\rightarrow\infty$. This
behavior is illustrated in Figure 10.

Figure 10: Numerical solutions for Eq. (48) by considering $\epsilon=0$ and
$\epsilon=\frac{r_{0}}{M_{\text{Cas}}}\left(2+\nu+M_{\text{Cas}}r_{0}\right)$
for $x_{0}=1$. Our calculations assumed $\nu=0.05$,
$M_{\text{Cas}}=1.38\sqrt{\sigma}$, and $\Phi_{0}=0$.

###  III.1 Source properties and stability analysis

At this point, we can determine all the characteristics associated with the
geometry and matter content of these structures. Remembering the shape
function of our Casimir-Yang-Mills wormhole solution:

$b(r)=r+2\kappa\lambda\sigma^{-\frac{\nu}{2}}M_{\text{Cas}}^{\nu+1}r\left[\Gamma(-\nu-1,M_{\text{Cas}}r)-\Gamma(-\nu-1,M_{\text{Cas}}r_{0})\right],$
(56)

and applying our deformation method (49), we found a redshift function to
arbitrary $n$ integer, regularized on the throat, given by:

$\Phi^{(n)}(r)=\int_{r_{0}}^{r}\frac{M_{\text{Cas}}^{-\nu-1}\bar{r}{}^{-\nu-n-2}\left[\bar{r}^{n}\left(2+\nu+M_{\text{Cas}}\bar{r}\right)-r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\right]e^{-M_{\text{Cas}}\bar{r}}}{2\left[\Gamma(-\nu-1,M_{\text{Cas}}\bar{r})-\Gamma(-\nu-1,M_{\text{Cas}}r_{0})\right]}d\bar{r},$
(57)

where we use
$\epsilon=\frac{r_{0}^{n}(2+\nu+M_{\text{Cas}}r_{0})}{M_{\text{Cas}}}$. With
the energy density coming from the Casimir energy for the $SU(N)$ gauge
theory:

$\rho(r)=-\lambda\frac{\left(r\sqrt{\sigma}\right)^{-\nu}}{r^{3}}e^{-M_{\text{Cas}}r},$
(58)

we derived the radial pressure using the standard relation adopted in the
usual Casimir effect:

$\displaystyle p_{r}^{(n)}(r)$ $\displaystyle=\omega_{r}^{(n)}(r)\rho(r),$
$\displaystyle=\left[2+\nu+M_{\text{Cas}}r-\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(\frac{r_{0}}{r}\right)^{n}\right]\rho(r).$
(59)

Note that, as in the early case, the mean value of the $\omega$-deformation is
null:

$\lim_{r\rightarrow\infty}\frac{1}{r-r_{0}}\int_{r_{0}}^{r}\left[\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(\frac{r_{0}}{r^{\prime}}\right)^{n}\right]dr^{\prime}=0.$
(60)

And, therefore, on average, again the method maintains the Casimir relations.
From Eq. (5), with the help Eqs. (56) and (57), the lateral pressure for the
Yang-Mills case assumes the form:

$\displaystyle p_{\theta}^{(n)}(r)$
$\displaystyle=\frac{\lambda\sigma^{-\frac{\nu}{2}}M_{\text{Cas}}^{-\nu-1}r^{-2\nu-2(n+2)}e^{-2M_{\text{Cas}}r}}{2\left[\Gamma\left(-\nu-1,M_{\text{Cas}}r\right)-\Gamma\left(-\nu-1,M_{\text{Cas}}r_{0}\right)\right]}$
$\displaystyle\times\left\\{\left(rr_{0}\right){}^{n}\left(5+2\nu+2M_{\text{Cas}}r\right)\left(2+\nu+M_{\text{Cas}}r_{0}\right)\right.$
$\displaystyle\
-r^{2n}\left(2+\nu+M_{\text{Cas}}r\right)\left(3+\nu+M_{\text{Cas}}r\right)$
$\displaystyle\ -r_{0}^{2n}\left(2+\nu+M_{\text{Cas}}r_{0}\right){}^{2}$
$\displaystyle\
-2r^{n}r_{0}^{-\nu-1}e^{M_{\text{Cas}}r}\left[r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(2+n+\nu+M_{\text{Cas}}r\right)\right.$
$\displaystyle\left.-r^{n}\left(4+4\nu+\nu^{2}+3M_{\text{Cas}}r+2\nu
M_{\text{Cas}}r+M_{\text{Cas}}^{2}r^{2}\right)\right]$ $\displaystyle\
\times\left.\left[r_{0}^{\nu+1}E_{\nu+2}(M_{\text{Cas}}r)-r^{\nu+1}E_{\nu+2}(M_{\text{Cas}}r_{0})\right]\right\\},$
(61)

where $E_{n}(z)=\int_{1}^{\infty}\frac{e^{-zt}}{t^{n}}dt$ gives the
exponential integral function. As expected, the EoM (3-5) and the conservation
law (6) are satisfied by the above expressions. Finally, we analyzed the
energy conditions and observed that, surprisingly, despite having different
relations from the noninteracting case, the interacting case exhibits
extremely similar graphical behavior, as evidenced in Fig. 11. Once again, we
have the violation of the NEC, DEC, and also of the SEC.

---
(a) (b)
(c) (d)
Figure 11:  Density and radial pressures (a), density and lateral pressures
(b), and combination of density and pressures (c) and (d) in order to verify
energy conditions, namely, DEC, considering $n=1,2,3$, $\sqrt{\sigma}=1$,
$\nu=0.05$, $M_{\text{Cas}}=1.38$, and $r_{0}=\lambda=1$, in Planck units.

####  III.1.1 Stability from sound velocity

The stability analysis of our solutions, based on the examination of sound
velocity within the source as defined by Eq. (30), demonstrates a consistent
alignment with the noninteracting case previously studied. Specifically, we
observe sustained stability near the wormhole throat, albeit with a
diminishing stability region as the power parameter $n$ increases. This
suggests an increasing susceptibility to instability as the solution
approaches the exact Casimir configuration, in line with the noninteracting
scenario. However, a departure from this latter is found in instances where
the sound velocity falls below the speed of light for certain parameter
values, a phenomenon absent in the noninteracting case. The visual
representation of this behavior is provided in Figure 12, illustrating the
parameter space region where $0\leq v_{s}^{2}\leq 1$, denoting stability and
physical validity.

Figure 12: Parameter space ($r_{0},M_{\text{Cas}}$) nearby the wormhole throat
($r\approx r_{0}$) for the interacting scenario, highlighting the region in
which $0\leq v_{s}^{2}\leq 1$, with $\nu=0.05$, $n=1$,
$\lambda=\frac{3\zeta(3)}{16\pi}$. The $\sigma$ parameter is constrained by
Eq. (46).

####  III.1.2 Stability from TOV equation

Similar to the previous case, the stability analysis for the non-abelian
version of the Casimir wormhole can be conducted with the help of the TOV
equation (32). By utilizing Eqs (57), (58), (59) and (61), we can precisely
determine the forces acting on the matter comprising our wormhole solution for
any positive integer value of $n$, specifically,

$\displaystyle\mathcal{F}_{h}$ $\displaystyle=$ $\displaystyle-\frac{\lambda
e^{-M_{\text{Cas}}r}}{\sigma^{\nu/2}r^{4+\nu}}$ (62)
$\displaystyle\times\left[6+\nu^{2}+\left(5+2M_{\text{Cas}}r\right)\nu+\left(4+M_{\text{Cas}}r\right)M_{\text{Cas}}r\right.$
$\displaystyle\left.-\left(\frac{r_{0}}{r}\right)^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(3+n+\nu+rM_{\text{Cas}}\right)\right],$
$\displaystyle\mathcal{F}_{g}$ $\displaystyle=$
$\displaystyle\frac{\lambda\left(\sqrt{\sigma}M_{\text{Cas}}\right)^{-\nu}\left[r^{n}\left(2+\nu+M_{\text{Cas}}r\right)-r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\right]e^{-2M_{\text{Cas}}r}}{2M_{\text{Cas}}r^{5+n+2\nu}\left[\Gamma\left(-\nu-1,M_{\text{Cas}}r\right)-\Gamma\left(-\nu-1,M_{\text{Cas}}r_{0}\right)\right]}$
(63)
$\displaystyle\times\left[3+\nu+M_{\text{Cas}}r-\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(\frac{r_{0}}{r}\right)^{n}\right],$

and

$\displaystyle\mathcal{F}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda\sigma^{-\frac{\nu}{2}}r^{-5-2(n+\nu)}e^{-2M_{\text{Cas}}r}$
(64)
$\displaystyle\times\left\\{2e^{M_{\text{Cas}}r}r^{1+n+\nu}\left[r^{n}\left(4+\nu^{2}+2\nu\left(2+M_{\text{Cas}}r\right)+M_{\text{Cas}}r\left(3+rM_{\text{Cas}}\right)\right)\right.\right.$
$\displaystyle\left.-r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\left(2+n+\nu+M_{\text{Cas}}r\right)\right]$
$\displaystyle+\frac{M_{\text{Cas}}^{-1-\nu}\left(r^{n}\left(2+\nu+M_{\text{Cas}}r\right)-r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)\right)}{\Gamma\left(-\nu-1,M_{\text{Cas}r}\right)-\Gamma\left(-\nu-1,M_{\text{Cas}}r_{0}\right)}$
$\displaystyle\times\left[r_{0}^{n}\left(2+\nu+M_{\text{Cas}}r_{0}\right)+2r^{n}e^{M_{\text{Cas}}r}E_{2+\nu}\left(M_{\text{Cas}}r\right)\right.$
$\displaystyle\left.\left.-r^{n}\left(3+\nu+M_{\text{Cas}}r+2e^{M_{\text{Cas}}r}E_{2+\nu}\left(M_{\text{Cas}}r_{0}\right)\left(\frac{r}{r_{0}}\right)^{1+\nu}\right)\right]\right\\}.$

Once again, we have verified that the equilibrium condition (34) is satisfied
for all $r$, assuming that the parameters of the model, such as $\lambda$,
$\sigma$, $\nu$, and $M_{\text{Cas}}$, are real and positive quantities (using
the computer algebra program Martin-Garcia:2008yei ). Under this criterion,
our solution represents a stable wormhole. Figure 13 demonstrates that the
equilibrium stage can be achieved due to the combined effect of pressure
hydrostatic, gravitational, and anisotropy forces. It is worth noting that the
value of $\mathcal{F}_{h}$ is positive, while $\mathcal{F}_{g}$ and
$\mathcal{F}_{a}$ are negative near the throat, resulting in a balanced system
when they are combined, resembling the behavior in the noninteracting case.

---
(a) (b)
Figure 13:  The graphical representation of $\mathcal{F}_{h}$,
$\mathcal{F}_{g}$, and $\mathcal{F}_{a}$ as functions of the radial coordinate
$r$, with $\sqrt{\sigma}=1$, $\nu=0.05$, $M_{\text{Cas}}=1.38$, and
$r_{0}=\lambda=1$, in Planck units.

##  IV Conclusion

In summary, our study has led to the discovery of three-dimensional
traversable wormhole solutions sourced by the Casimir density and pressure
related to the quantum vacuum fluctuations of the Yang-Mills field. We have
initiated the analysis at the tree level (from the noninteracting Casimir
energy) with an arbitrary state parameter of $\omega$. The determination of
the throat radius through the requirement of asymptotic flatness is an
important feature of these solutions, and we have found that it is on the
order of $r_{0}\sim\tilde{G}$ (in natural units). The constant shape function
equal to this radius further highlights the elegance of those solutions.

Our novel methodology of deforming the state parameter $\omega$ by adding a
small function obeying an inverse power law of the radial coordinate has
allowed us to find finite redshift functions throughout all space. This
methodology is critical to understanding the wormhole’s behavior since,
without it, the object behaves as a black hole for $\omega<0$ and is not a
wormhole for other values of this parameter. Additionally, our analysis
reveals that the space-averaged deformation function approaches zero,
indicating that the object behaves as a legitimate Casimir wormhole with an
expected average state parameter of $\omega=2$. This behavior becomes more
pronounced as the power parameter $n$ increases.

Our study has also revealed valuable insights into the curvature properties of
the obtained family of wormhole solutions. Specifically, we have found that
the Kretschmann scalar reaches a global maximum at the wormhole’s throat and
increases with the power parameter $n$. We have also observed a local maximum
in the Kretschmann scalar that increases with $n$. Although the local minimum
of the Kretschmann scalar shifts slightly towards the throat with increasing
$n$, it remains on the curve corresponding to $n=1$.

We have also examined the null energy condition (NEC) and the dominant energy
condition (DEC) and found that these conditions are violated throughout all
space. The strong energy condition (SEC) becomes not satisfied since the NEC
is broken. These results emphasize the need for further investigation into the
stability and viability of these solutions in physical scenarios.

Next, we examined the stability of this specific family of Casimir wormholes,
focusing on the squared sound velocity in the surrounding medium. The results
demonstrated that the obtained traversable solutions are stable near the
wormhole throat, and the size of the stable region diminishes as the parameter
$n$ increases. However, these noninteracting solutions were found to be
unphysical since the sound speed consistently exceeded the speed of light near
the wormhole throat, irrespective of the selected parameter values. In the
context of the TOV equation, our analysis confirmed the system’s stability
across all spatial domains.

We also have discovered a novel family of traversable wormhole solutions,
utilizing the quantum vacuum fluctuations of interacting Yang-Mills fields as
a source. Through analytical derivation of the shape function, we have
confirmed its compliance with all the requirements for generating a
traversable wormhole. We have successfully obtained a well-behaved redshift
function by modifying the effective state parameter, following a similar
approach as in our previous examination. These wormhole solutions exhibit
sensitivity to system parameters such as the radius of the wormhole throat,
Casimir mass, anomalous exponent, and string tension. Notably, our research
has revealed that higher string tension results in a larger throat radius,
which can be attributed to the stretching effect in an attempt to deconfine
the gluons.

In the sequel, we also investigated the stability of interacting Y-M Casimir
wormholes based on squared sound velocity within the source. We found stable
solutions near the wormhole throat, with diminishing stability as they
approached the Casimir configuration (for increasing $n$ powers), mirroring
the noninteracting case. Notably, certain parameter values led to sound
velocity dropping below the speed of light, differentiating these solutions
and enhancing their physical relevance. Thus, in the context of the non-
interacting scenario, subluminal velocities are observed alongside stability
only within a limited region around the wormhole throat. In this case, on
considering the other space regions, as the power parameter $n$ increases,
superluminal velocities, and instability become more pronounced, similar to
the increase in energy condition violations, as we can compare Fig. 2 with
Fig. 3. On the other hand, in the interacting scenario, stability and
subluminal velocities coexist across the entire space, depending on specific
parameter choices. Notably, this coexistence persists irrespective of energy
condition violations, as previously demonstrated. Moreover, considering the
TOV equation revealed consistent stability across all space, akin to the prior
case.

In conclusion, our results reveal new possibilities for constructing Casimir
wormhole solutions and contribute to our understanding of their properties.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grants no. 88887.822058/2023-00 (ACLS), no. 308268/2021-6
(CRM), and no. 200879/2022-7 (RVM) for financial support. R. V. Maluf thanks
the Department of Theoretical Physics $\&$ IFIC of the University of Valencia
- CSIC for the kind hospitality. The authors also thank the anonymous referee
for his valuable comments and suggestions.

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# Direct Measurement of Dipoles in Anomalous Elasticity of Amorphous Solids

Bhanu Prasad Bhowmik  Dept. of Chemical Physics, The Weizmann Institute of
Science, Rehovot 76100, Israel  Michael Moshe  Racah Institute of Physics, The
Hebrew University of Jerusalem, Jerusalem, Israel 9190  Itamar Procaccia∗
Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100,
Israel, ∗Center for OPTical IMagery Analysis and Learning, Northwestern
Polytechnical University, Xi’an, 710072 China

###### Abstract

Recent progress in studying the physics of amorphous solids has revealed that
mechanical strains can be strongly screened by the formation of plastic events
that are typically quadrupolar in nature. The theory stipulated that gradients
in the density of the quadrupoles act as emergent dipole sources, leading to
strong screening and to qualitative changes in the mechanical response, as
seen for example in the displacement field. In this Letter we firstly offer
direct measurements of the dipole field, independently of any theoretical
assumptions, and secondly we demonstrate detailed agreement with the recently
proposed theory. These two goals are achieved using data from both simulations
and experiments. Finally we show how measurements of the dipole fields
pinpoint the theory parameters that determine the profile of the displacement
field.

Introduction: Classical elasticity theory is one of the basic building blocks
of Condensed Matter Physics. Once the elastic moduli of a given material are
known [1], one can predict the displacement field that is associated with any
mechanical strain [2], using linear elasticity when the latter is small, and
applying nonlinear extensions of the theory when the strains are larger [3].
This happy state of affairs is however questioned when one studies the
mechanical properties of amorphous solids. One reason is that in amorphous
solids plastic responses appear instantaneously for any amount of strain [4].
Another reason is that nonlinear elastic constants were shown to have
unbounded sample to sample fluctuations in the thermodynamic limit [5].
Finally, at least in frictional amorphous solids, one observes stress
correlations that are not consistent with classical elasticity theory [6, 7].
It is thus necessary to revisit elasticity theory in the context of amorphous
solids with the aim of seeking an applicable theory that can provide
predictive power.

Such a theory was presented in a recent paper [8]. Attention was given to the
plastic responses, which typically appear as quadrupolar (Eshelby-like)
irreversible responses [9, 10, 11]. When the density of these quadrupoles is
low, they act only to renormalize the elastic moduli, but they do not change
the form of the theory. This is reminiscent the role of dipoles in
dielectrics, where the dielectric constant is dressed, but the structure of
electrostatic theory remains intact [12]. On the other hand, when the density
of quadrupoles is high, the gradients of their density cannot be ignored, and
these are acting as dipoles. Dipole-dipole and dipole-displacement interaction
become crucial, and these change the structure of the theory and the resulting
mechanical responses.

Brief review of the theory: To introduce the theory in its simplest form we
focus here on 2-dimensional systems with radial geometry. Specifically, we
consider amorphous configurations of binary disks, with and without friction.
For the frictionless case we examined in numerical simulations the
displacement field that results from an inflation of a disk closest to the
center. For the frictional case our collaborators studied experimentally the
displacement field that is induced by a pusher at the center of coordinates
whose effect is similar to the inflation considered in silico. Details of
these simulations and experiments can be found in Refs. [8, 13]. In short, in
our experiment and simulations we consider an annulus of radii $r_{\text{in}}$
and $r_{\text{out}}$, $r_{\text{in}}\ll r_{\text{out}}$, with an imposed
displacement $\mathbf{d}(r_{\text{in}})=d_{0}\hat{r}$ and
$\mathbf{d}(r_{\text{out}})=0$. The polar symmetry of the problem implies that
$\mathbf{d}(r)=d_{r}(r)\hat{r}$, in which case normal elasticity theory
implies the equation

$\Delta{\mathbf{d}}=d_{r}^{\prime\prime}+\frac{d_{r}^{\prime}}{r}-\frac{d_{r}}{r^{2}}=0\
.$ (1)

The solution to this differential equation that satisfies the boundary
conditions is

${d}_{r}(r)=d_{0}\frac{r^{2}-r_{\text{out}}^{2}}{r_{\text{in}}^{2}-r_{\text{out}}^{2}}\frac{r_{\text{in}}}{r}\
.$ (2)

One should note that the solution (2) is a positive monotonically decreasing
function of $r$, and it decays at large values of $r$ like $1/r$, as is
expected in standard elasticity theory. It was shown [8] that this form of the
displacement field is expected to remain valid also when there exists a low
(or uniform) density of plastic events, although the coefficients may get
renormalized. We then refer to this situation as “quasi-elastic”.

The situation changes qualitatively when the density of plastic events gains
sizeable gradients, and these act as dipole sources [8]. The screening caused
by dipoles changes Eq. (2) to read

$d_{r}^{\prime\prime}+\frac{d_{r}^{\prime}}{r}-\frac{d_{r}}{r^{2}}=-\kappa^{2}d_{r}\,$
(3)

with $\kappa$ being an emergent constant that is not known a-priori. Below we
refer to $\kappa$ as the “screening parameter”. Eq. (3) is equivalent to the
Bessel equation whose solution, satisfying $d_{r}(r_{\rm in})=d_{0}$,
$d_{r}(r_{\rm out})=0$, reads

$d_{r}(r)=d_{0}\frac{Y_{1}(r\,\kappa)J_{1}(r_{\text{out}}\kappa)-J_{1}(r\,\kappa)Y_{1}(r_{\text{out}}\kappa)}{Y_{1}(r_{\text{in}}\kappa)J_{1}(r_{\text{out}}\kappa)-J_{1}(r_{\text{in}}\kappa)Y_{1}(r_{\text{out}}\kappa)}\
.$ (4)

Here $J_{1}$ and $Y_{1}$ are the Bessel functions of the first and second kind
respectively. For very small values of $\kappa$ the solution reduces back to
(2). Importantly, depending on the precise value of $\kappa$, Eq. (4) may non
monotonic, negative, and even oscillatory. We refer to situations that agree
with the solution Eq. (4) and are either non-monotonic or oscillating as
“anomalous elasticity”. Until now the numerical values of $\kappa$ had to be
determined by fitting Eq. (4) to experiments and simulations [8, 13]. One
consequence of the present Letter will be a direct determination of the
$\kappa$ as shown below.

Two examples for further analysis: Our next aim is to establish the existence
(or nonexistence) of dipole fields, directly from the measured displacement
field. We choose two examples, the first being a numerical simulation of
frictionless binary amorphous assembly of disks [8], and the second an
experiment with frictional binary disks [13], all in circular geometry, and
both showing anomalous responses. The displacement fields in these cases
result from an inflation of a disk or a pusher at the center of coordinates.
The angle averaged radial component of the displacement field as measured in a
simulation and an experiment respectively are shown in Fig. 1.

Figure 1: Angle averaged radial components of the displacement field as
measured in simulation and experiment, cf. Refs. [8, 13]. Panel (a):
simulations with frictionless Hertzian disks, exhibiting an anomalous
response. The continuous line is the solution Eq. (4) with $r_{\rm in}=4.8$,
$r_{\rm out}=72.2$ and $\kappa=0.0525$. Panel (b): experimental measurements
with frictional disks, The continuous line is the solution Eq. (4) with
$r_{\rm in}=133.3$, $r_{\rm out}=3478$ and $\kappa=0.00147$.

Both examples deviate strongly from the elastic solution (2). Rather, the
continuous lines in both panels represent the anomalous solutions (4) with
$\kappa$ being a fitting parameter. Panels (a) and (b) are obtained with
$\kappa=0.0525$ and $\kappa=0.00147$ respectively. We note here that the
systems are of different sizes and $\kappa$ is dimensional. For the sake of
comparison one should consider a dimensionless number
$\tilde{\kappa}\equiv\kappa r_{\rm out}$. Then the two examples have
$\tilde{\kappa}=3.79$ and 5.11 respectively. One of our goals below is to
develop a direct measure for the screening parameter $\kappa$ and to test its
success in predicting the curves in Fig. 1, without fitting.

Theory-independent measurement of dipoles: We firstly consider the existence
of dipole sources in our measured displacement field, independently of our
proposed theory. To this aim we adopt the stress function formalism in which
screening charges are introduced as sources for the bi-harmonic equation for
the Airy function [14]. Distributed dipoles, either induced or fixed, are
incorporated into the equation as a source term [15, 16]

$\frac{1}{Y}\Delta\Delta\chi=\nabla\cdot\mathbf{P}$ (5)

where $Y$ is the Young modulus and $\mathbf{P}(x)$ is the position dependent
dipole density.

Remembering that the pressure $p$ is given by $p\equiv\Delta\chi$, we now
consider the integral of the LHS of Eq. 5 over a domain $\Omega$:

$\begin{split}\frac{1}{Y}\int_{\Omega}\Delta\Delta\chi\text{dS}=\frac{1}{Y}\oint_{\partial\Omega}(\nabla\Delta\chi)\cdot\mathbf{n}\text{dl}=\frac{1}{Y}\oint_{\partial\Omega}(\nabla
p)\cdot\mathbf{n}\,\text{dl}\end{split}$ (6)

where ${\bm{n}}$ is the outgoing unit vector on the boundary of the domain. On
the other hand, the integral of the RHS of Eq. (5) reads

$\int_{\Omega}\nabla\cdot\mathbf{P}\text{dS}=\oint_{\partial\Omega}\mathbf{P}\cdot\mathbf{n}\,\text{dl}\
.$ (7)

We therefore conclude that the signature for the existence of non-uniform
dipole distribution is from the integral over the pressure flux

$\begin{split}\frac{1}{Y}\oint_{\partial\Omega}(\nabla
p)\cdot\mathbf{n}\,\text{dl}=\oint_{\partial\Omega}\mathbf{P}\cdot\mathbf{n}\,\text{dl}\;.\end{split}$
(8)

Note that the pressure is proportional to the trace of the strain
$p=Y\,\mathrm{Tr}(u)$, which in turn describes the volume deformation of the
material and is equal to the displacement divergence
$\mathrm{Tr}(u)=\nabla\cdot\mathrm{d}$. Denoting the resulting integral as
$I_{1}$ we end up with

$\oint_{\partial\Omega}\mathbf{P}\cdot\mathbf{n}\,\text{dl}=\oint_{\partial\Omega}(\nabla(\nabla\cdot\mathrm{d}))\cdot\mathbf{n}\,\text{dl}\equiv
I_{1}\;.$ (9)

In the case of our circular systems with radial symmetry both sides of this
equation can be evaluated analytically. The RHS reads

$\oint_{\partial\Omega}(\nabla(\nabla\cdot\mathrm{d}))\cdot\mathbf{n}\,\text{dl}=2\pi
r\Big{(}d_{r}^{{}^{\prime\prime}}(r)+\frac{d_{r}^{\prime}(r)}{r}-\frac{d_{r}(r)}{r^{2}}\Big{)}\
,$ (10)

and the LHS reads

$\oint_{\partial\Omega}\mathbf{P}\cdot\mathbf{n}=2\pi rP(r)\ ,$ (11)

hence

$P(r)=d_{r}^{{}^{\prime\prime}}(r)+\frac{d_{r}^{\prime}(r)}{r}-\frac{d_{r}(r)}{r^{2}}\
.$ (12)

We should notice that Eq. (1) guarantees that Eq. (10) is zero identically
when classical elasticity is obeyed. Since this is correct for every closed
circle, it means that ${\bm{P}}(r)=0$. This is consistent with the absence of
dipole sources. Note also that in 2-dimensions the elasticity theory solution
for the radial component of the displacement field, being proportional to
$1/r$ at large values of $r$ results in a trivial zero on the RHS of Eq. (10).

At this point we return to the two examples shown in Fig. 1, and compute the
dipole density in Eq. (12), directly from the measured radial component. The
results for the frictionless simulation and the frictional experiment are
shown in Fig. 2.

Figure 2: The integral $I_{1}$ of Eq. (10) computed for the two anomalous
examples shown in panels (a) and (b) of Fig. 1. These plots use the data
without any resort to theoretical fits. The existence of these integrals are a
direct demonstration for the presence of dipole fields.  Figure 3: The same
integral $I_{1}$ of Eq. (10) computed for the two examples shown in Fig. 1.
Here we use the fit Eq.(4) for the sake of smoothness. These integrals are
identical to those obtained using Eq. (15) up to the scale $\kappa^{2}$ as is
demanded from the solution Eq. (4).

Although the data is somewhat noisy, we can see that the measured dipole
density is non-zero for the two cases, indicating that indeed dipole sources
exist in the system, as they emerge together with the onset of the
displacement fields. We stress that this result is independent of the theory
proposed in Ref. [8]. To obtain a smoother measurement we use the fit of the
data to Eq. (4). The result of this smoother evaluation is shown for the two
examples in Fig. 3. The two methods of evaluation are consistent with each
other and show that a dipole field is emerging together with the displacement
field which is caused by the inflation near the center of our amorphous solids
in their radial geometry.

Theory dependent evaluation: Next we test the theory that was proposed in Ref.
[8] where an anomalous response of the system is described by Eq. (3)

$\nabla\mathrm{Tr}(u)=(\nabla(\nabla\cdot\mathrm{d}))=-\kappa^{2}\mathbf{d}$
(13)

hence the integrand in LHS of Eq. (10) can be simplified to

$\oint_{\partial\Omega}\mathbf{P}\cdot\mathbf{n}\,\text{dl}=-\kappa^{2}\oint_{\partial\Omega}\mathbf{d}\cdot\mathbf{n}\,\text{dl}\
.$ (14)

If the theory is correct, the evaluation using the RHS of Eq. (14) should
agree with the evaluation of Eq. (10). Denote the integral itself as $I_{2}$,

$I_{2}\equiv-\oint_{\partial\Omega}\mathbf{d}\cdot\mathbf{n}\,\text{dl}\ .$
(15)

Since we have used in Fig. 3 the solution of Eq. (4) to compute $I_{1}$, it is
obvious that $I_{2}$ would be identical to $I_{1}$ up to the scale factor
$\kappa^{2}$. Computing $\sqrt{I_{1}/I_{2}}$ one indeed finds the evident
value of $\kappa$ which was used to fit the data in Fig. 1.

Figure 4: Comparison of the solutions obtained using the screening parameter
$\kappa$ from the measurements of the dipole fields. Panel (a): comparison to
the experimental data (blue points) as shown in Fig. 1 panel (a). Panel (b)
comparison to the simulation data (blue points) shown in panel (b) of Fig. 1.

Thus so far we did not put the theory of Ref. [8] under a stringent test. To
achieve such a test we return to the raw data of the angle averaged
displacements shown in Fig. 1. To avoid effects of noise we smooth the
displacement data by calculating moving averages. The displacement at each
position $r$ is estimated from the average of 8 nearest neighbors. In
addition, we note that the integrals $I_{1,2}$ measure the dipole density
$P(r)$. Upon integrating over the radial direction we find the total dipole
charge induced in the system

$\begin{split}\mathcal{P}_{\mathrm{tot}}&=2\pi\int r\,P(r)\mathrm{d}r\\\
&=\kappa^{2}\int I_{2}(r)\mathrm{d}r=\int I_{1}(r)\mathrm{d}r\end{split}$ (16)

Therefore we can estimate $\kappa$ by

$\begin{split}\kappa=\sqrt{\frac{\int I_{1}(r)\mathrm{d}r}{\int
I_{2}(r)\mathrm{d}r}}\ .\end{split}$ (17)

This ratio can be calculated directly from the raw data. We find the values
$\kappa=0.04980$ and $\kappa=0.00146$ for the two examples in Fig. 1. If the
theory is correct, when these measured calues of $\kappa$ are plugged in to
the predicted form in Eq. (4) the resulting curves should agree with the
measured radial components of the displacement without performing any fitting
procedure. Indeed, in Fig. 4 we plot the predicted form (4) and the measured
displacement using the calculated $\kappa$ and find very good agreement, thus
providing a support to the dipole screening picture in general and to our
specific model in particular.

Summary and conclusions: The aim of this Letter was twofold. Firstly we wanted
to present a direct computation of the dipole field that was presumed to exist
in Refs. [8, 13]. Contrary to the case of electrostatics [12], where dipoles
can be directly attributed to polarized molecules or polymers, in the present
case dipoles were identified as gradients of quadrupole fields, rendering them
less obvious. We have shown in this Letter how to compute the dipole fields
from loop integrals on the displacement field. The existence of a dipole field
goes hand in hand with a finite value of the screening parameter $\kappa$ and
with having anomalous elasticity. Mutatis mutandis, when $\kappa=0$ we expect
quasi-elastic responses with a possible renormalization of the elastic moduli.

The second aim of this Letter was to present a method to estimate the
numerical value of the screening parameter $\kappa$ from the measurement of
the dipole field. The total dipole charge, as defined in Eq. (16), provides us
with a method to compute $\kappa$ from Eq. (17), without fitting the angle-
averaged displacement field. The agreement of the prediction of Eq. (4) with
the data, once we used the screening parameter from this procedure, is a
strong support for the theory as presented in Refs. [8, 13].

It should be stated that until now the theory was tested by and compared to
simulations and experiments involving disks in radial geometry in
2-dimensions. In the near future we will provide evidence that the theory is
also applicable to systems of spheres in 3-dimensions, and to standard models
of glass formers in 2 and 3-dimensions.

acknowledgments: We thank Chandana Mondal for helpful discussions and sharing
data. This work had been supported in part by Minerva Foundation and the
Minerva Center for “Aging, from physical materials to human tissues” at the
Weizmann Institute. MM acknowledges support from the Israel Science Foundation
(grant No. 1441/19).

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