# A scalar matching factor on the Birkhoff polytope characterizing permutation
and uniform matrices

Suvadip Sana111Email<EMAIL_ADDRESS>
Department of Statistics and Data Science, Cornell University

###### Abstract

Birkhoff polytope is the set of all bistochastic matrices (also known as
doubly stochastic matrices). Bistochastic matrices form a special class of
stochastic matrices where each row and column sums up to one. Permutation
matrices and uniform matrices are special extreme cases of bistochastic
matrices. In this paper, we define a scalar quantity called the matching
factor on the Birkhoff polytope. Given a bistochastic matrix, we define the
matching factor by taking the product of the squares of the Euclidean norms of
each row and column and show that permutation matrices and uniform matrices
maximize and minimize the matching factor, respectively. We also extend this
definition of scalar matching factor to a larger class of matrices and show
similar maximization and minimization properties.

##  1 Introduction

Birkhoff polytope is defined as the set of all bistochastic matrices.
Bistochastic matrices appear in various fields of study, such as the theory of
Markov chains [7][8], the theory of Majorization [4], and several physical
problems [2]. Random bistochastic matrices have also been considered [3].
Additionally, they are important in the theory of optimization and matching
[9][6]. Permutation matrices are a special class of bistochastic matrices, as
they form the building blocks of bistochastic matrices according to the
celebrated Birkhoff-Von Neumann theorem [1]. This theorem states that every
bistochastic matrix can be written as a convex combination of permutation
matrices. In the literature, there are a few characterizations of permutation
matrices [5]. Permutation matrices are also useful in describing perfect
matching in bipartite graphs. Another important class of bistochastic matrices
is uniform matrices, which appear as the limiting distribution of a Markov
chain whose transition matrix is a bistochastic matrix. In the Markov chain
literature, permutation transition matrices refer to completely deterministic
chains, whereas uniform chains refer to uniform randomization over the state
space of the Markov chains. An interesting question to ask is whether there is
a simple scalar quantity (which is easy to compute) that characterizes these
two extreme cases. So far, to the best of our knowledge, no such
characterization has been proposed. There have been a few works on the
characterization of stochastic matrices, see e.g., [11]. In this paper, we
contribute to the literature by defining a single scalar quantity on the
Birkhoff polytope whose maximum and minimum values characterize permutation
and uniform matrices, respectively. We organize the paper as follows: in
section 2, we give all the definitions and notations to be used in this paper;
in section 3, we state and prove the main results; in section 4, we conclude
with certain remarks.

##  2 Definitions and Notations

Notations : $\mathbb{R},\mathbb{N}$ will denote the set of all real numbers
and natural numbers respectively. We will call a square matrix $A$ of
dimension $n\times n$ to be a matrix of order $n$. For a matrix $A$, $A_{k.}$
(note the dot after $k$) will denote the $k$th row of the matrix and $A_{.k}$
(note the dot before $k$) will denote the $k$-th column of the matrix. we call
a matrix $A$ to be non-negative if all the entries are non-negative.
$\|.\|_{2}^{2}$ will denote the squared euclidean norm.

for $n\in\mathbb{N}$, we define six types of matrices of order $n$ below,

######  Definition 1.

We say a non-negative matrix $B$ of order $n$ to be a bistochastic matrix if

$\sum_{i=1}^{n}B_{ki}=\sum_{j=1}^{n}B_{jk}=1,\ \ k\in\\{1,\dots,n\\}$

Some examples of bistochastic matrices are

$\begin{bmatrix}1/2&1/2\\\ 1/2&1/2\end{bmatrix}\ ,\
\begin{bmatrix}1/3&1/3&1/3\\\ 0&2/3&1/3\\\ 2/3&0&1/3\end{bmatrix}$

We denote the set of all bistochastic matrices of order $n$ as
$\mathcal{B}_{n}$ (Birkhoff polytope)

######  Definition 2.

We say a bistochastic matrix $P$ of order $n$ to be a permutation matrix if
each row and column has exactly only one-zero entry and that non-zero entry is
equal to one.

Some examples of permutation matrices are

$\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\ ,\ \begin{bmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{bmatrix}$

We denote the set of all permutation matrices of order $n$ by
$\mathcal{P}_{n}$

######  Definition 3.

We say a bistochastic matrix $U$ of order $n$ to be a uniform matrix if

$U_{ij}=\frac{1}{n},\ \ \ i,j\in\\{1,\dots,n\\}$

Some examples of uniform matrices are

$\begin{bmatrix}1/2&1/2\\\ 1/2&1/2\end{bmatrix}\ ,\
\begin{bmatrix}1/3&1/3&1/3\\\ 1/3&1/3&1/3\\\ 1/3&1/3&1/3\end{bmatrix}$

Clearly, there is only one uniform matrix of order $n$, we denote that uniform
matrix by $U_{n}$.

Next we state three more definitions that generalizes above three standard
definitions

######  Definition 4.

We say a non-negative matrix $A$ to be a $*$-positive matrix if each row and
column contain at least one positive entry

Note that any bistochastic matrix is a $*$\- positive matrix and some other
examples of $*$-positive matrix are

$\begin{bmatrix}1/2&0\\\ 1/2&1/2\end{bmatrix}\ ,\ \begin{bmatrix}1/3&0&0\\\
1/3&7/8&0\\\ 1/2&0&1/3\end{bmatrix}$

We denote the set of all $*$-positive matrices of order $n$ as
$\mathcal{B}^{*}_{n}$

######  Definition 5.

We say a non-negative matrix $A$ to be a $*$-permutation matrix if each row
and column contain exactly one positive entry

Note that any permutation matrix is a $*$\- permutation matrix some other
examples of $*$-permutation matrix are

$\begin{bmatrix}1/2&0\\\ 0&1/32\end{bmatrix}\ ,\ \begin{bmatrix}1/7&0&0\\\
0&0&5\\\ 0&1/5&0\end{bmatrix}$

We denote the set of all $*$-permutation matrices of order $n$ as
$\mathcal{P}^{*}_{n}$

######  Definition 6.

We say any $*$\- positive matrix $A$ to be a $*$-uniform matrix if all the
entries are equal.

Note that any uniform matrix is a $*$\- uniform matrix, some other examples of
$*$-uniform matrix are

$\begin{bmatrix}1/6&1/6\\\ 1/6&1/6\end{bmatrix}\ ,\ \begin{bmatrix}1&1&1\\\
1&1&1\\\ 1&1&1\end{bmatrix}$

We denote the set of all $*$-positive matrices of order $n$ as
$\mathcal{U}^{*}_{n}$

##  3 Main result

Before we state and prove our main characterization theorem, we first state an
important observation to be used in the proof.

######  Lemma 1.

if $a_{1},\dots,a_{n}$ are $n$ non-negative real numbers, then only one of
them is equal to one and rest are zero if and only if

$\sum_{k=1}^{n}a_{k}^{2}=(\sum_{k=1}^{n}a_{k})^{2}=1$ (3.1)

###### Proof.

Note that 3.1 implies after expanding the square

$\sum_{i<j}a_{i}a_{j}=0$

And the above holds if and only if at most one of them is non-zero, but we
also have that $\sum_{k=1}^{n}a_{k}^{2}=1$, and all $a_{1},\dots,a_{n}$ are
non-negative, hence exactly one of them is equal to one and rest are zero.
Other direction is trivial. ∎

Next we define some quantities for stating our main result. For a bistochastic
matrix $B$ of order $n$ let $B_{1.},\dots B_{n.}$ denote the rows of $B$ and
let $B_{.1},\dots B_{.n}$ denote the columns of $B$. For each $k$-th row and
$k$-column, we define $k$-th Matching factor of $B$ as follows

$\lambda(k)=\|B_{k.}\|_{2}^{2}\|B_{.k}\|_{2}^{2}\ \ \ \ k=1,\dots,n$

Intuitively if $\lambda(k)$ is near one, then one of the entry is nearly one
in each row and column and rest are near zero, where as if $\lambda(k)$ in
small, then many entries can be non-zero. The reason behind this intuition is
the geometry of the $l_{2}$ ball and $l_{1}$ ball in $\mathbb{R}^{n}$, see
[10]. we gave the name Matching factor because the corresponding row and
column matches to exactly one non-zero entry when $\lambda$ is equal to one.

For a bistochastic matrix $B$, we denote the Matching factor of $B$ as $M(B)$
and define it as follows

$M(B)=\prod_{k=1}^{n}\lambda(k)$

Intuitively the quantity $M(B)$ captures the entire matching factor of the
matrix by combined contribution of the matching factor of each row and column.
Next we state our characterization theorem.

######  Theorem 1.

for any bistochastic matrix $B$ of order $n$, the matching factor $M(B)$
satisfies the inequalities

$\frac{1}{n^{2n}}\leq M(B)\leq 1$

and the lower bound and upper bounds are achieved

$\max_{B\in\mathcal{B}_{n}}M(B)=1\ \ \
\operatorname*{arg\,max}M(B)=\mathcal{P}_{n}$ (3.2)

and

$\min_{B\in\mathcal{B}_{n}}M(B)=\frac{1}{n^{2n}}\ \ \
\operatorname*{arg\,min}M(B)=\\{\mathcal{U}_{n}\\}$ (3.3)

###### Proof.

First we prove 3.2. Note that if $B$ is permutation matrix, then by definition
each row and column of $B$ will have exactly non-zero entry which is one and
rest zero, hence we will get that

$\|B_{.k}\|_{2}^{2}=\|B_{k.}\|_{2}^{2}=1\ \ \ \forall\ \ k\in\\{1,\dots,n\\}$

.

Hence we get $M(B)=1$.

Now to prove the other direction. suppose we have $M(B)=1$.

By now non-negativity of each entries of $B$, and by the sum constraints we
must have

$\displaystyle\|B_{.k}\|_{2}^{2}=\sum_{i=1}^{n}B_{ik}^{2}$
$\displaystyle\leq(\sum_{i=1}^{n}B_{ik})^{2}=1\ \ \ \forall\ \
k\in\\{1,\dots,n\\}$ (3.4)
$\displaystyle\|B_{k.}\|_{2}^{2}=\sum_{i=1}^{n}B_{ki}^{2}$
$\displaystyle\leq(\sum_{i=1}^{n}B_{ki})^{2}=1\ \ \ \forall\ \
k\in\\{1,\dots,n\\}$ (3.5)

Hence for general bistochastic matrix we must have that $\lambda(k)\leq 1\
\forall\ k\in\\{1,\dots,n\\}$, which implies $M(B)\leq 1$.

Now from our assumption we have $M(B)=1$. Hence we must have $\lambda(k)=1\
\forall\ k\in\ \ \\{1,\dots,n\\}$, which in turn implies that all inequalites
in 3.4 and 3.5 must be equalities. Hence now by using lemma 1, we get each row
and each column of $B$ must have exactly non-zero entry equal to one and rest
zero which in turn implies that $B$ must be a permutation matrix, hence we are
done.

Now we prove 3.3. The proof follows from the well known Cauchy-Schwarz
inequality.

Note that by Cauchy-Schwarz inequality

$\displaystyle\|B_{.k}\|^{2}=\sum_{i=1}^{n}B_{ik}^{2}$
$\displaystyle\geq\frac{1}{n}(\sum_{i=1}^{n}B_{ik})^{2}=\frac{1}{n}\ \
\forall\ k\in\ \ \\{1,\dots,n\\}$ (3.7)
$\displaystyle\|B_{k.}\|^{2}=\sum_{j=1}^{n}B_{kj}^{2}$
$\displaystyle\geq\frac{1}{n}(\sum_{j=1}^{n}B_{jk})^{2}=\frac{1}{n}\ \
\forall\ k\in\ \ \\{1,\dots,n\\}$ (3.8)

Hence from 3.7 and 3.8, we will get that $\lambda(k)\geq\frac{1}{n^{2}}\ \
\forall k\in\\{1,\dots,n\\}$

Hence we must get that for any bistochastic matrix,
$M(B)=\prod_{k=1}^{n}\lambda(k)\geq\frac{1}{n^{2n}}$

and note that the equality $M(B)=\frac{1}{n^{2n}}$ holds if and only if all
the inequalities in 3.7 and 3.8 are equalities and by the conditions of
equality in cauchzy schwarz inequality, we must have that all entries should
be equal and since each row and column sum upto one, we must have
$B_{ij}=\frac{1}{n}\ \ \forall\ i,j\in\\{1,\dots,n\\}.$. Hence the minimum is
achieved only at the uniform matrix. Hence we have established our result.

∎

Next we state and prove similar results involving $*$\- positive matrix, $*$\-
permutation matrix and $*$-uniform matrix.

we again define similar quanties for $*$\- positive matrix. For a $*$\-
positive matrix $A$ of order $n$ let $A_{1.},\dots A_{n.}$ denote the rows of
$A$ and let $A_{.1},\dots A_{.n}$ denote the columns of $A$. For each $k$-th
row and $k$-th column, we define $k$-th Matching factor of $A$ as follows

$\lambda^{*}(k)=\frac{\|A_{k.}\|_{2}^{2}\|A_{.k}\|_{2}^{2}}{(\sum_{i=1}^{n}A_{ki})^{2}(\sum_{j=1}^{n}A_{jk})^{2}}\
\ \ \ k=1,\dots,n$

Intuitively if $\lambda(k)$ is near one, then one of the entry is nearly one
and rest are near zero, where as if $\lambda(k)$ in small, then many entries
can be non-zero.

For a $*$-positive matrix, we denote the Matching factor of $A$ as $M^{*}(A)$
and define it as follows

$M^{*}(A)=\prod_{k=1}^{n}\lambda^{*}(k)$

Note that above definitions of Matching factor reduces to the same definition
of Matching factor for bistochastic matrices. Next we state our
characterization theorem for $*$-positive matrices.

######  Theorem 2.

for any $*$-positive matrix $A$ of order $n$, the matching factor $M^{*}(A)$
satisfies the inequalities

$\frac{1}{n^{2n}}\leq M^{*}(A)\leq 1$

and the lower bound and upper bounds are achieved

$\max_{A\in\mathcal{A}^{*}_{n}}M^{*}(A)=1\ \ \
\operatorname*{arg\,max}M(A)=\mathcal{P}^{*}_{n}$ (3.9)

and

$\min_{A\in\mathcal{A}^{*}_{n}}M^{*}(A)=\frac{1}{n^{2n}}\ \ \
\operatorname*{arg\,min}M^{*}(A)=\mathcal{U}^{*}_{n}\ $ (3.10)

###### Proof.

First we prove 3.9. Note that if $A$ is $*$-permutation matrix, then by
definition each row and column of $A$ will have exactly non-zero entry and
rest zero, hence we will get that

$\displaystyle\|A_{.k}\|_{2}^{2}=\sum_{i=1}^{n}A_{ik}^{2}$
$\displaystyle=(\sum_{i=1}^{n}A_{ik})^{2}\ \ \forall\ \ k\in\\{1,\dots,n\\}$
$\displaystyle\|A_{k.}\|_{2}^{2}=\sum_{i=1}^{n}A_{ki}^{2}$
$\displaystyle=(\sum_{i=1}^{n}A_{ki})^{2}\ \ \forall\ \ k\in\\{1,\dots,n\\}$

Hence by combining, we get

$\lambda^{*}(k)=\frac{\|A_{.k}\|_{2}^{2}\|A_{k.}\|_{2}^{2}}{(\sum_{i=1}^{n}A_{ki})^{2}(\sum_{j=1}^{n}A_{jk})^{2}}=1\
\ \ \forall\ \ k\in\\{1,\dots,n\\}$

.

Hence we get $M^{*}(A)=1$.

Now to prove the other direction. suppose we have $M^{*}(A)=1$.

By non-negativity of each entries of $A$, and that each row and column has
atleast one positive entry, and from the trivial observation that for any non-
negative numbers $a_{1},\dots,a_{n}$ we must have
$\sum_{k=1}^{n}a^{2}_{k}\leq(\sum_{k=1}^{n}a_{k})^{2}$, so we get

$\frac{\|A_{.k}\|_{2}^{2}\|A_{k.}\|_{2}^{2}}{(\sum_{i=1}^{n}A_{ki})^{2}(\sum_{j=1}^{n}A_{jk})^{2}}\leq
1\ \ \ \forall\ \ k\in\\{1,\dots,n\\}$ (3.11)

Hence for any $*$\- positive matrix, we must have that $\lambda^{*}(k)\leq 1\
\forall k\ \in\\{1,\dots,n\\}$, which implies $M^{*}(A)\leq 1$.

Now from our assumption we have $M^{*}(A)=1$. Hence we must have
$\lambda^{*}(k)=1\ \forall k\in\ \ \\{1,\dots,n\\}$ Which in turn implies that
all inequalites in 3.11 must be equalities, So we will have

$\displaystyle\sum_{i=1}^{n}A_{ik}^{2}$
$\displaystyle=(\sum_{i=1}^{n}A_{ik})^{2}\ \ \forall\ \ k\in\\{1,\dots,n\\}$
$\displaystyle\sum_{i=1}^{n}A_{ki}^{2}$
$\displaystyle=(\sum_{i=1}^{n}A_{ki})^{2}\ \ \forall\ \ k\in\\{1,\dots,n\\}$

Now we state a slight modification of lemma 1 below

Modification of lemma 1: if $a_{1},\dots,a_{n}$ are $n$ non-negative real
numbers, then only exactly one of them is non-zero if and only if

$\sum_{k=1}^{n}a_{k}^{2}=(\sum_{k=1}^{n}a_{k})^{2}>0$ (3.12)

The proof of the above modification is a trivial modification of the proof of
lemma 1. So we get each row and each column of $A$ must have exactly one non-
zero entry and rest are zero which in turn implies that $A$ must be a $*$\-
permutation matrix, hence we are done.

Now we prove 3.10. The proof again follows from Cauchy-Schwarz inequality.

Note that by Cauchy-Schwarz inequality

$\frac{\|A_{.k}\|_{2}^{2}\|A_{k.}\|_{2}^{2}}{(\sum_{i=1}^{n}A_{ki})^{2}(\sum_{j=1}^{n}A_{jk})^{2}}\geq\frac{1}{n^{2}}\
\ \ \forall\ \ k\in\\{1,\dots,n\\}$ (3.13)

Hence from 3.13, we will get that $\lambda^{*}(k)\geq\frac{1}{n^{2}}\ \
\forall k\in\\{1,\dots,n\\}$

Hence we must get that for any $*$\- positive matrix,
$M^{*}(A)=\prod_{k=1}^{n}\lambda^{*}(k)\geq\frac{1}{n^{2n}}$

and note that the equality $M^{*}(A)=\frac{1}{n^{2n}}$ holds if and only if
all the inequalities in 3.13 are equalities and by the conditions of equality
in cauchzy schwarz inequality, we must have that all entries should be equal
Hence the minimum is achieved only for the $*$-uniform matrix. Hence we have
established our result.

∎

##  4 Conclusion and remarks

We defined a scalar matching factor on a bistochastic matrix and showed that
the lower and upper bounds characterize the permutation matrices and uniform
matrices. We also defined similar quantities for a larger class of matrices,
known as $*$-positive matrices, and demonstrated similar maximization and
minimization properties. The scalar matching factor on bistochastic matrices
can be of independent interest in optimization, matching, and the theory of
Markov chains. One possible application could be checking or validating
whether a large bistochastic matrix is a permutation matrix by simply
computing our matching factor rather than checking row by row. This matching
quantity seems to provide a notion of how far any bistochastic matrix is from
being a permutation matrix, though more thought will be needed to correctly
formulate such notions.

##  5 Acknowledgements

The Author acknowledge the summer support from the Department of Statistics
and Data science at Cornell university. Special thanks to Suraj Dash of UT
Austin for taking part in discussion of some of the ideas presented in the
paper.

## References

  * [1] Birkhoff, G. Three observations on linear algebra.  Univ. Nac. Tacuman, Rev. Ser. A 5 (1946), 147–151.
  * [2] Brualdi, R. A. Combinatorial matrix classes, vol. 13.  Cambridge University Press, 2006.
  * [3] Cappellini, V., Sommers, H.-J., Bruzda, W., and Życzkowski, K. Random bistochastic matrices.  Journal of Physics A: Mathematical and Theoretical 42, 36 (2009), 365209.
  * [4] Cohen, J., Kempermann, J. H., and Zbaganu, G. Comparisons of stochastic matrices with applications in information theory, statistics, economics and population.  Springer Science & Business Media, 1998.
  * [5] Cruse, A. B. A proof of fulkerson’s characterization of permutation matrices.  Linear Algebra and its Applications 12, 1 (1975), 21–28.
  * [6] Linderman, S. W., Mena, G. E., Cooper, H., Paninski, L., and Cunningham, J. P. Reparameterizing the birkhoff polytope for variational permutation inference, 2017.
  * [7] Maksimov, V. M. Convergence of non-homogeneous bistochastic markov chains.  Theory of Probability & Its Applications 15, 4 (1970), 604–618.
  * [8] Mukhamedov, F., and Embong, A. F. On stable b-bistochastic quadratic stochastic operators and associated non-homogenous markov chains.  Linear and Multilinear Algebra 66, 1 (2018), 1–21.
  * [9] Munkres, J. Algorithms for the assignment and transportation problems.  Journal of the society for industrial and applied mathematics 5, 1 (1957), 32–38.
  * [10] Tibshirani, R. Regression shrinkage and selection via the lasso.  Journal of the Royal Statistical Society Series B: Statistical Methodology 58, 1 (1996), 267–288.
  * [11] von Below, J., and Rénier, S. A characterization of even doubly-stochastic matrices.  Discrete Mathematics 309, 6 (2009), 1273–1283.


In our setting, we do not have access to the SQ oracle with finely labeled
examples. Our main result of this section (Theorem 3) is a mechanism that
enables us to obtain access to such an oracle using a few coarsely labeled
examples (with high probability). Hence, we can still perform the noisy
gradient descent of the previous paragraph with an additional overhead on the
sample complexity, due to the reduction.

##  Appendix B Multiclass Logistic Regression with Coarsely Labeled Data

A first application for the above generic reduction from coarse data to
statistical queries is the case of coarse multiclass logistic regression. In
the standard (finely labeled) multiclass logistic regression problem, there
are $k$ fine labels (that correspond to classes), each one associated with a
weight vector $\bm{w}_{z}\in\mathbb{R}^{n}$ with $z\in[k]$. We can consider
the weight matrix $\bm{W}\in\mathbb{R}^{k\times n}$. Given an example
$\bm{x}\in\mathbb{R}^{n}$, the vector $\bm{x}$ is filtered via the softmax
function $\sigma(\bm{W},\bm{x})$, which is a probability distribution over
$\Delta^{k}$ with
$\sigma(\bm{W},\bm{x};z)=\exp(\bm{w}_{z}^{T}\bm{x})/\sum_{y\in[k]}\exp(\bm{w}_{y}^{T}\bm{x}),z\in[k]$
and the output is the finely labeled example
$(\bm{x},z)\in\mathbb{R}^{n}\times[k]$. The goal is to estimate the weight
matrix $\bm{W}$, given finely labeled examples. Let us denote by
$\mathcal{D}(\bm{W})$ the joint distribution over the finely labeled examples
for simplicity. When we have access to finely labeled examples
$(\bm{x},z)\sim\mathcal{D}(\bm{W}^{\star})$, the population log-likelihood
objective $\mathcal{L}$ of the multiclass logistic regression problem

$\mathcal{L}(\bm{W})=\operatorname*{\mathbf{E}}_{(\bm{x},z)\sim\mathcal{D}({\bm{W}^{\star}})}\Big{[}\bm{w}_{z}^{T}\bm{x}-\log\Big{(}\sum_{j\in\mathcal{Z}}\exp(\bm{w}_{j}^{T}\bm{x})\Big{)}\Big{]}\,,$

is concave (see [FHT+01]) with respect to the weight matrix
$\bm{W}\in\mathbb{R}^{k\times n}$ and is solved using gradient methods. On the
other hand, if we have sample access only to coarsely labeled examples
$(\bm{x},S)\sim\mathcal{D}_{\pi}(\bm{W}^{\star})$, the population log-
likelihood objective $\mathcal{L}_{\pi}$ of the coarse multiclass logistic
regression problem

$\mathcal{L}_{\pi}(\bm{W})=\operatorname*{\mathbf{E}}_{(\bm{x},S)\sim\mathcal{D}_{\pi}(\bm{W}^{\star})}\Big{[}\log\Big{(}\sum_{z\in
S}\exp(\bm{w}_{z}^{T}\bm{x})\Big{)}-\log\Big{(}\sum_{j\in\mathcal{Z}}\exp(\bm{w}_{j}^{T}\bm{x})\Big{)}\Big{]}\,,$

which is no more concave. However, as an application of our main result
(Theorem 3), we can still solve it. In fact, since we can implement
statistical queries using the sample access to the coarse data generative
process $\mathcal{D}_{\pi}(\bm{W}^{\star})$, we can compute the gradients of
the log-likelihood objective that corresponds to the _finely labeled
examples_. Hence, the total sample complexity of optimizing this non-convex
objective is equal to the sample complexity of solving the convex problem with
an additional overhead at each iteration of computing the gradients, that is
given by Theorem 3.

##  Appendix C Geometric Information Preservation

In this section, we aim to provide some intuition behind the notion of
information preserving partitions. The following result provides a geometric
property for the partition distribution $\pi$. We show that if the partition
distribution satisfies this particular geometric property, then it is also
information preserving. We underline that the geometric property is quite
important for our better understanding and it has the advantage that it is
easy to verify. Hence, while the notion of information preserving
distributions may be less intuitive, we believe that the geometric
preservation property that we state in Lemma 19 can fulfill this lack of
intuition. The property informally states that, for any hyperplane, the sets
in the partition that are not cut by this hyperplane have non trivial
probability mass with respect to the true Gaussian. In the case of mixtures of
convex partitions, we would like the same property to hold in expectation.

Figure 4:  (a) is a very rough partition that makes learning the mean
impossible: Gaussians $\mathcal{N}((0,z))$ centered along the same vertical
line $(0,z)$ assign exactly the same probability to all cells of the
partitions and therefore,
$\mathrm{TV}(\mathcal{N}_{\pi}((0,z_{1})),\mathcal{N}_{\pi}((0,z_{2})))=0$: it
is impossible to learn the second coordinate of the mean. (b) is a convex
partition of $\mathbb{R}^{2}$, that makes recovering the Gaussian possible.

Before stating Lemma 19, let us return to Figure 4. Observe that, in the first
example with the four halfspaces, the geometric property does not hold, since
there exists a line (i.e., a hyperplane) that intersects with all the sets. On
the other hand, if we consider the second example with the Voronoi partition
and assume that the true mean lies in the middle of the picture, we can see
that any hyperplane does not intersect with a sufficient number of sets and,
hence, the union of the uncut sets has non trivial probability mass for any
hyperplane.

For a hyperplane
$\mathcal{H}_{\bm{w},c}=\\{\bm{x}\in\mathbb{R}^{d}:\bm{w}^{T}\bm{x}=c\\}$ with
normal vector $\bm{w}\in\mathbb{R}^{d}$ and threshold $c\in\mathbb{R}$, we
denote the two associated halfspaces by
$\mathcal{H}_{\bm{w},c}^{+}=\\{\bm{x}\in\mathbb{R}^{d}:\bm{w}^{T}\bm{x}>c\\}$
and
$\mathcal{H}_{\bm{w},c}^{-}=\\{\bm{x}\in\mathbb{R}^{d}:\bm{w}^{T}\bm{x}<c\\}$.
Before stating the next Lemma, we shortly describe what means for a hyperplane
to cut a set with respect to a Gaussian $\mathcal{N}$. The set $S$ is not cut
by the hyperplane $\mathcal{H}$, if it totally lies in a halfspace induced by
the hyperplane, say $\mathcal{H}^{+}$, i.e., it holds that
$\mathcal{N}(S)=\mathcal{N}(S\cap\mathcal{H}^{+})$.

######  Lemma 19 (Geometric Information Preservation).

Consider the generative process of coarse $d$-dimensional Gaussian data
$\mathcal{N}_{\pi}(\bm{\mu}^{\star}),$ (see Definition 4). Consider an
arbitrary hyperplane $\mathcal{H}_{\bm{w},c}$ with normal vector
$\bm{w}\in\mathbb{R}^{d}$ and threshold $c\in\mathbb{R}$. For a partition
$\mathcal{S}\in\mathrm{supp}(\pi)$ of $\mathbb{R}^{d}$, consider the
collection that contains all the sets that are not cut by the hyperplane
$\mathcal{H}_{\bm{w},c}$, i.e.,

$U_{\bm{w},c,\mathcal{S}}=\bigcup\Big{\\{}S\in\mathcal{S}:\mathcal{N}^{\star}(S\cap\mathcal{H}_{\bm{w},c}^{+})=\mathcal{N}^{\star}(S)\lor\mathcal{N}^{\star}(S\cap\mathcal{H}_{\bm{w},c}^{-})=\mathcal{N}^{\star}(S)\Big{\\}}\,.$

Assume that $\pi$ satisfies

$\operatorname*{\mathbf{E}}_{\mathcal{S}\sim\pi}\Big{[}\mathcal{N}(\bm{\mu}^{\star};U_{\bm{w},c,\mathcal{S}})\Big{]}\geq\alpha\,,$
(C.1)

for some $\alpha\in(0,1]$. Then, for any Gaussian distribution
$\mathcal{N}(\bm{\mu})$, it holds that

$\mathrm{TV}(\mathcal{N}_{\pi}(\bm{\mu}),\mathcal{N}_{\pi}(\bm{\mu}^{\star}))\geq
C_{\alpha}\cdot\mathrm{TV}(\mathcal{N}(\bm{\mu}),\mathcal{N}(\bm{\mu}^{\star}))\,,$

for some $C_{\alpha}$ that depends only on $\alpha$ and satisfies
$C_{\alpha}=\Omega(\alpha^{3})$, i.e., the partition distribution is
$C_{\alpha}$-information preserving.

Hence, the geometric property is sufficient for information preservation. If
we assume that the total variation distance between the true Gaussian
distribution $\mathcal{N}(\bm{\mu}^{\star})$ and a possible guess
$\mathcal{N}(\bm{\mu})$ is at least $\epsilon$ and the partition distribution
satisfies the geometric property of Equation (C.1), we get that the coarse
generative process preserves a sufficiently large gap, in the sense that
$\mathrm{TV}(\mathcal{N}_{\pi}(\bm{\mu}^{\star}),\mathcal{N}_{\pi}(\bm{\mu}))\geq\mathrm{poly}(\alpha)\epsilon$.
The proof of the above lemma, which relies on high-dimensional anti-
concentration results on Gaussian distributions, follows.

###### Proof of Lemma 19.

Let us denote the true distribution by
$\mathcal{N}^{\star}=\mathcal{N}(\bm{\mu}^{\star},\bm{I})$ for short. Consider
an arbitrary hyperplane $\mathcal{H}_{\bm{w},c}$ with normal vector
$\bm{w}\in\mathbb{R}^{d}$ and threshold $c\in\mathbb{R}$. Since the partition
distribution (supported on a family of partitions $\mathcal{B}$) satisfies
Equation (C.1), we have that, for the random variable
$\mathcal{N}^{\star}(U_{\bm{w},c,\mathcal{S}}),$ that takes values in $[0,1]$,
there exists $\alpha$ such that

$\operatorname*{\mathbf{E}}_{\mathcal{S}\sim\pi}\Big{[}\mathcal{N}^{\star}(U_{\bm{w},c,\mathcal{S}})\Big{]}=\alpha\,.$

We will use the following simple Markov-type inequality for bounded random
variables.

######  Fact 1 (Lemma B.1 from [SSBD14]).

Let $Z$ be a random variable that takes values in $[0,1]$. Then, for any
$\alpha\in(0,1)$, it holds that

$\operatorname*{\mathbf{Pr}}[Z>\alpha]\geq\frac{\operatorname*{\mathbf{E}}[Z]-\alpha}{1-\alpha}\geq\operatorname*{\mathbf{E}}[Z]-\alpha\,.$

By the 1, it holds that

$\operatorname*{\mathbf{Pr}}_{\mathcal{S}\sim\pi}\Big{[}\mathcal{N}^{\star}(U_{\bm{w},c,\mathcal{S}})\geq\alpha/2\Big{]}\geq\alpha/2\,.$

Fix such a partition $\mathcal{S}\in\mathcal{B}$ (in the effective support of
the partition distribution) and consider the true
$\mathcal{N}^{\star}=\mathcal{N}(\bm{\mu}^{\star})$ and the guess
$\mathcal{N}=\mathcal{N}(\bm{\mu})$ distributions. For this pair of
distributions, consider the set

$\mathcal{H}=\Big{\\{}\bm{x}\in\mathbb{R}^{d}:\bm{x}^{T}(\bm{\mu}-\bm{\mu}^{\star})=\big{(}\|\bm{\mu}\|_{2}^{2}-\|\bm{\mu}^{\star}\|_{2}^{2}\big{)}/2\Big{\\}}\,.$

Observe that this set is a hyperplane with normal vector
$\bm{\mu}^{\star}-\bm{\mu}$, that contains the midpoint
$\frac{1}{2}(\bm{\mu}+\bm{\mu}^{\star})$. Our main focus is to lower bound the
total variation distance of the coarse distributions
$\mathcal{N}^{\star}_{\pi}$ and $\mathcal{N}_{\pi}$. We claim that this lower
bound can be described as a fractional knapsack problem and, hence, it is
attained by a worst-case set, that (intuitively) places points as close as
possible to the hyperplane $\mathcal{H}$, until its mass with respect to the
true Gaussian $\mathcal{N}^{\star}$ is at least $\alpha/2$. Recall that the
total variation distance between the two coarse distributions is

$\mathrm{TV}(\mathcal{N}_{\pi},\mathcal{N}^{\star}_{\pi})=\sum_{\mathcal{S}\in\mathcal{B}}\pi(\mathcal{S})\sum_{S\in\mathcal{S}}\Big{|}\mathcal{N}(S)-\mathcal{N}^{\star}(S)\Big{|}\,.$

The following optimization problem gives a lower bound on the total variation
distance between $\mathcal{N}^{\star}_{\pi}$ and $\mathcal{N}_{\pi}$.

$\min_{S}\Big{|}\int(\mathcal{N}(\bm{\mu}^{\star};\bm{x})-\mathcal{N}(\bm{\mu};\bm{x}))\bm{1}_{S}(\bm{x})d\bm{x}\Big{|}\,,$
$\text{subj.
to}~{}~{}~{}\int\mathcal{N}(\bm{\mu}^{\star};\bm{x})\bm{1}_{S}(\bm{x})d\bm{x}\geq\alpha/2\,.$

We begin with a claim about the shape of the worst case set. Let
$t=(\|\bm{\mu}\|_{2}^{2}-\|\bm{\mu}^{\star}\|_{2}^{2})/2$ be the hyperplane
threshold.

$\bm{\mu}_{1}=\bm{\mu}^{*}$$\bm{\mu}_{2}$$\mathcal{H}$ Figure 5: Illustration
of the worst-case set in testing the hypotheses
$h_{1}=\\{\bm{\mu}_{1}=\bm{\mu}^{\star}\\}$ and
$h_{2}=\\{\bm{\mu}_{2}=\bm{\mu}^{\star}\\}.$

######  Claim 6.

Let $\mathcal{H}^{+}=\\{\bm{x}:\bm{x}^{T}(\bm{\mu}-\bm{\mu}^{\star})<t\\}$ and
$\mathcal{H}^{-}=\\{\bm{x}:\bm{x}^{T}(\bm{\mu}-\bm{\mu}^{\star})>t\\}$. The
mass of the solution of the fractional knapsack is totally contained in either
$\mathcal{H}^{+}$ or $\mathcal{H}^{-}$.

Since the partition distribution satisfies Equation (C.1) with respect to the
true Gaussian $\mathcal{N}(\bm{\mu}^{\star})$ and since the set $\mathcal{H}$
is a hyperplane, the probability mass that is not cut by $\mathcal{H}$ is at
least $\alpha$. Hence, there exists a halfspace (either $\mathcal{H}^{+}$ or
$\mathcal{H}^{-}$) with mass at least $\alpha/2$. Recall that any partition
contains only convex sets. Also, observe that the hyperplane $\mathcal{H}$ is
also the the zero locus of the the polynomial
$q(\bm{x})=\|\bm{x}-\bm{\mu}\|_{2}^{2}-\|\bm{x}-\bm{\mu}^{\star}\|_{2}^{2}$
and, hence, it is the set of points where the two spherical Gaussians
$\mathcal{N}(\bm{\mu})$ and $\mathcal{N}(\bm{\mu}^{\star})$ assign equal mass.
Hence, we have that

$\mathcal{H}^{+}=\Big{\\{}\bm{x}:\mathcal{N}(\bm{\mu}^{\star})>\mathcal{N}(\bm{\mu})\Big{\\}}\,.$

Hence, we can assume that the worst-case set lies (totally) in
$\mathcal{H}^{+}$ and, then, the optimization problem can be written as

$\min_{S}\int\Big{(}1-\frac{\mathcal{N}(\bm{\mu};\bm{x})}{\mathcal{N}(\bm{0};\bm{x})}\Big{)}\mathcal{N}(\bm{0};\bm{x})\bm{1}_{S}(\bm{x})d\bm{x}\,,$
$\text{subj.
to}~{}~{}~{}\int\mathcal{N}(\bm{0};\bm{x})\bm{1}_{S}(\bm{x})d\bm{x}\geq\alpha/2,~{}~{}S\in\mathcal{H}^{+}\,.$

Without loss of generality, we assume that
$\mathcal{N}^{\star}=\mathcal{N}(\bm{0},\bm{I})$ and
$\mathcal{N}=\mathcal{N}(\bm{\mu},\bm{I})$. In order to design the worst-case
set, since the optimization has the structure of the fractional knapsack
problem, we can think of each point $\bm{x}\in\mathcal{H}^{+}$ as having
_weight_ equal to its contribution to the mass gap
$(\mathcal{N}(\bm{0};\bm{x})-\mathcal{N}(\bm{\mu};\bm{x}))$ and _value_ equal
to its density with respect to the true Gaussian $\mathcal{N}(\bm{0};\bm{x})$.
Hence, in order to design the worst-case set, the points
$\bm{x}\in\mathcal{H}^{+}$ should be included in the set in order of
increasing ratio of weight over value, until reaching a threshold $T$. So, we
can define the worst-case set to be

$S=\Big{\\{}\bm{x}\in\mathcal{H}^{+}:1-\frac{\mathcal{N}(\bm{\mu};\bm{x})}{\mathcal{N}(\bm{0};\bm{x})}\leq
T\Big{\\}}=\Big{\\{}\bm{x}\in\mathcal{H}^{+}:1-\exp(p(\bm{x}))\leq
T\Big{\\}}\,,$

where
$p(\bm{x})=-\frac{1}{2}(\bm{\mu}-\bm{x})^{T}(\bm{\mu}-\bm{x})+\frac{1}{2}\bm{x}^{T}\bm{x}=-\frac{1}{2}\bm{\mu}^{T}\bm{\mu}+\bm{\mu}^{T}\bm{x}$
and note that $p(\bm{x})\leq 0$ for any $\bm{x}\in\mathcal{H}^{+}$. We will
use the following anti-concentration result about the Gaussian mass of sets,
defined by polynomials.

######  Lemma 20 (Theorem 8 of [CW01]).

Let $q,\gamma\in\mathbb{R}_{+},\bm{\mu}\in\mathbb{R}^{d}$ and $\bm{\Sigma}$ in
the positive semidefinite cone $\mathbb{S}^{d}_{+}$. Consider
$p:\mathbb{R}^{d}\rightarrow\mathbb{R}$ a multivariate polynomial of degree at
most $\ell$ and let

$\mathcal{Q}=\Big{\\{}\bm{x}\in\mathbb{R}^{d}:|p(\bm{x})|\leq\gamma\Big{\\}}\,.$

Then, there exists an absolute constant $C$ such that

$\mathcal{N}(\bm{\mu},\bm{\Sigma};\mathcal{Q})\leq\frac{Cq\gamma^{1/\ell}}{(\operatorname*{\mathbf{E}}_{\bm{z}\sim\mathcal{N}(\bm{\mu},\bm{\Sigma})}[|p(\bm{z})|^{q/\ell}])^{1/q}}\,.$

We can apply Lemma 20 for the quadratic polynomial $p(\bm{x})$ by setting
$\gamma=\frac{\alpha^{2}}{256C^{2}}\sqrt{\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}[p^{2}(\bm{x})]}$
with $q=4$, where $C$ is the absolute Carbery-Wright constant. Hence, we get
that the Gaussian mass of the set
$\mathcal{Q}=\\{\bm{x}:|p(\bm{x})|\leq\gamma\\}$ is equal to

$\mathcal{N}^{\star}(\mathcal{Q})\leq\alpha/4\,.$

Hence, for any point $\bm{x}$ in the remaining $\alpha/4$ mass of the set $S$,
it holds that $|p(\bm{x})|\geq\gamma$. We first observe observe that $\gamma$
can lower bounded by the total variation distance of $\mathcal{N}^{\star}$ and
$\mathcal{N}$. It suffices to lower bound the expectation
$\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}[p^{2}(\bm{x})]$.
We have that

$\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}\Big{[}p^{2}(\bm{x})\Big{]}\geq\operatorname*{\mathbf{V}}_{\bm{x}\sim\mathcal{N}^{\star}}\Big{[}p(x)\Big{]}=\operatorname*{\mathbf{V}}_{\bm{x}\sim\mathcal{N}^{\star}}\Big{[}-\frac{1}{2}\bm{\mu}^{T}\bm{\mu}+\bm{\mu}^{T}\bm{x}\Big{]}=\|\bm{\mu}\|_{2}^{2}\,,$

and, hence

$\gamma\geq\frac{\alpha^{2}}{256C^{2}}\|\bm{\mu}\|_{2}\,.$

We will use the following lemma for the total variation distance of two Normal
distributions.

######  Lemma 21.

Let
$N_{1}=\mathcal{N}(\bm{\mu}_{1},\bm{\Sigma}_{1}),N_{2}=\mathcal{N}(\bm{\mu}_{2},\bm{\Sigma}_{2})$
be two Normal distributions. Then, it holds

$\mathrm{TV}(N_{1},N_{2})\leq\frac{1}{2}\|\bm{\Sigma}_{1}^{-1/2}(\bm{\mu}_{1}-\bm{\mu}_{2})\|_{2}+\sqrt{2}\|\bm{I}-\bm{\Sigma}_{1}^{-1/2}\bm{\Sigma}_{2}\bm{\Sigma}_{1}^{-1/2}\|_{F}\,.$

Applying Lemma 21 to the above inequality, we get

$\gamma\geq\frac{\alpha^{2}}{256C^{2}}\mathrm{TV}(\mathcal{N}(\bm{\mu}),\mathcal{N}(\bm{\mu}^{\star}))\,.$

To conclude, we have to lower bound the $L_{1}$ gap between
$\mathcal{N}(\bm{0},\bm{I};\bm{x})\bm{1}_{S}(\bm{x})$ and
$\mathcal{N}(\bm{\mu},\bm{I};\bm{x})\bm{1}_{S}(\bm{x})$ and since $S$ lies
totally in $\mathcal{H}^{+}$

$\int_{S}(\mathcal{N}(\bm{0};\bm{x})-\mathcal{N}(\bm{\mu};\bm{x}))d\bm{x}=\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}\left[1-\exp(p(\bm{x}))\Big{|}\bm{1}_{S}(\bm{x})\right]\,.$

To proceed with distinguish two cases: First, assume that $\gamma\leq 1$ and
recall that $\mathcal{Q}=\\{\bm{x}:|p(\bm{x})|\leq\gamma\\}$. Note that for
$y\in[-1,0]$, it holds that $1-\exp(y)\geq|y|/2$ and, hence, we have that:

$\int_{S}(\mathcal{N}(\bm{0};\bm{x})-\mathcal{N}(\bm{\mu};\bm{x}))d\bm{x}\geq\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}\left[\frac{|p(\bm{x})|}{2}\bm{1}_{S\setminus\mathcal{Q}}(\bm{x})\right]\geq\gamma\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}\left[\bm{1}_{S\setminus\mathcal{Q}}(\bm{x})\right]\geq\frac{\alpha\gamma}{4}\,,$

and, by the lower bound for $\gamma$, we get

$\int_{S}(\mathcal{N}(\bm{0},\bm{I};\bm{x})-\mathcal{N}(\bm{\mu},\bm{I};\bm{x}))d\bm{x}\geq
C_{\alpha}\mathrm{TV}(\mathcal{N}(\bm{\mu}),\mathcal{N}(\bm{\mu}^{\star}))\,,$

for some $C_{\alpha}=\Omega(\alpha^{3})$. Otherwise, let $\gamma>1$. Note that
for $y<-1$, it holds that $1-\exp(y)\geq 1/2.$ Hence, we get that

$\int_{S}(\mathcal{N}(\bm{0};\bm{x})-\mathcal{N}(\bm{\mu};\bm{x}))d\bm{x}\geq\operatorname*{\mathbf{E}}_{\bm{x}\sim\mathcal{N}^{\star}}\left[\frac{1}{2}\bm{1}_{S\setminus\mathcal{Q}}(\bm{x})\right]\geq\alpha/8\,.$

In conclusion, we get that

$\mathrm{TV}(\mathcal{N}_{\pi}^{*},\mathcal{N}_{\pi})\geq
C_{\alpha}\mathrm{TV}(\mathcal{N}^{\star},\mathcal{N})\,.$

where $C_{\alpha}=\mathrm{poly}(\alpha)$ and depends only on $\alpha$. ∎


# Meson Potential Energy in a Non-Conformal Holographic Model

M<EMAIL_ADDRESS>School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5746,
Tehran, Iran  Ali<EMAIL_ADDRESS>School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O.Box
19395-5746, Tehran, Iran  Department of Physics, Shahid Beheshti University
G.C., Evin, Tehran 19839, Iran

###### Abstract

We study the meson potential energy in a non-conformal model at both zero and
finite temperature via gauge/gravity duality. This model consists of five-
dimensional Einstein gravity coupled to a scalar field with a non-trivial
potential. Interestingly, at both zero and finite temperature we find that the
relative meson potential energy can be considered as a measure of non-
conformality of the theory. At zero temperature we show that parameters of the
Cornell potential, i.e. Coulomb strength parameter $\kappa$ and constant $C$
depends on the energy scale $\Lambda$ that breaks conformal symmetry and the
difference between the number of degrees of freedom of UV and IR fixed points
$\Delta N$ while QCD string tension $\sigma_{s}$ just depends on the
$\Lambda$. At finite temperature we see that there is a melting length
$l_{m\ell}$ where beyond that the meson dissociates in the plasma and by
increasing $\Lambda$ the value of $l_{m\ell}$ increases while its value
decreases by increasing the temperature.

##  I Introduction

The Anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence or more
generally gauge/gravity duality provides a wide range of domain to study
strongly coupled quantum field theories whose duals are the gravitational
theories in one higher dimension Maldacena:1997re ; Witten:1998qj ;
Gubser:1998bc ; CasalderreySolana:2011us ; Ammon:2015wua . In fact, this
correspondence is a strong-weak duality that is a novel approach to attack the
problems of strongly coupled field theories using the classical gravity dual
which is a weakly coupled theory. Accordingly, parameters, fields, and
different phenomena in the gauge theory are translated into equivalents on the
gravity side. Therefore, many different questions in the strongly coupled
gauge theory can be set up and investigated in dual gravity. This conjectured
duality used to explore applications in the strongly coupled region of Quantum
Chromodynamics (QCD), i.e. holographic QCD Kruczenski:2003be ;
Kruczenski:2003uq ; Kobayashi:2006sb ; Babington:2003vm ; Sakai:2004cn ;
Sakai:2005yt ; Erlich:2005qh ; Karch:2006pv ; Li:2011hp ; DeWolfe:2010he ;
DeWolfe:2011ts ; Hajilou:2021wmz ; Amiri-Sharifi:2016uso ; Ali-Akbari:2015bha
; Bohra:2020qom ; Dudal:2021jav ; AliAkbari:2012vt ; Ali-Akbari:2013txa ;
Fang:2015ytf ; Callebaut:2011ab ; Arefeva:2020vae ; Braga:2017bml ;
Abt:2019tas ; Nakas:2020hyo .

One of the important strongly coupled systems is Quark-Gluon Plasma (QGP)
produced at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider
(LHC) by colliding heavy nuclei at a relativistic speed Shuryak:2003xe ;
Shuryak:2004cy . The potential energy between a quark and an anti-quark living
in the plasma is very useful for investigating the strong interactions and
quark confinement. One of the most important non-local and gauge invariant
observables in gauge theories is Wilson loop. In quantum field theory,
utilizing this gauge invariant and non-local quantity one can obtain the
potential energy Schwartz:2014sze . The expectation value of Wilson loop has
much power to tackle the interesting questions in QCD such as confinement
Wilson:1974sk . Concisely, our approach to calculate the static potential
energy of the heavy quark-antiquark pair is to compute the expectation value
of a rectangular Wilson loop in the strongly coupled plasma. It is important
to note that extracting the meson potential energy from Wilson operator in
holography is only valid in the heavy quark limit. For more details refer to
CasalderreySolana:2011us . The holographic dual of the rectangular Wilson loop
is given by a classical open string suspended from two points in the boundary
of the gravity theory and hanging down in the gravity bulk. In fact, Wilson
loop was used to calculate the potential energy of the heavy quark-antiquark
for the first time Maldacena:1998im . Using this idea, the meson potential
energy has been extensively investigated in static backgrounds
Brandhuber:1998bs ; Andreev:2006ct ; Yang:2015aia  and also is generalized to
time-dependent backgrounds Ali-Akbari:2015ooa ; Hajilou:2017sxf ;
Ishii:2014paa ; Hajilou:2018dcb .

Gauge/gravity duality covers the conformal theories as well as non-conformal
ones. It is very interesting to study the effect of non-conformality on the
physical quantities. For instance, the equilibration rates of strongly coupled
non-conformal quark-gluon plasmas is studied in Buchel:2015saa  and the
behavior of the potential energy and holographic subregion complexity
corresponding to a probe meson in a non-conformal background is investigated
in Lezgi:2020bkc . Also, by considering the non-conformal plasma the behavior
of the lowest nonhydrodynamic modes in a class of holographic models is
investigated in Janik:2015waa  and the dependence of the approach to thermal
equilibrium of strongly coupled non-conformal plasmas is analytically
determined in Gursoy:2015nza . In addition, sound wave propagation in strongly
coupled non-conformal gauge theory plasma is investigated in Benincasa:2005iv
.

In this paper in order to study the potential energy of heavy quark-antiquark
in a non-conformal model in a holographic set up we consider a five-
dimensional Einstein gravity coupled to a scalar field with a non-trivial
potential which is introduced and studied in Attems:2016ugt . In this model,
the corresponding gauge theory is a non-conformal field theory that has two
conformal fixed points at ultraviolet (UV) and at infrared (IR). In the
gravity side, these solutions are asymptotically AdS5 in the IR and UV limits
with different radii. Thermodynamics transport and relaxation properties of
this non-conformal background are studied in Attems:2016ugt . Also this
background utilized for studying entanglement entropy Rahimi:2016bbv , mutual
and tripartite information Ali-Akbari:2019zkf  and subregion complexity
Asadi:2020gzl .

The remainder of this paper is organized as follows. In Section II we review
on the non-conformal background and its holographic dual in the gauge theory
side at zero and finite temperature. In Section III we introduce the
rectangular Wilson loop and we then calculate the potential energy between a
quark and an antiquark in the plasma. In Section IV we explain our numerical
results and describe how parameters of the gauge theory and in particular the
energy scale can affect the meson potential energy. Finally, in Section V we
discuss our main results.

##  II Review on the backgrounds

We are interested in studying the meson potential energy of a holographic
five-dimensional model whose dual four-dimensional gauge theory is not
conformal Attems:2016ugt .

###  II.1 Zero Temperature

The non-conformal holographic model which we consider here consists of five-
dimensional dimensional Einstein gravity coupled to a scalar field with non-
trivial potential. The action of this model is given by:

$S=\frac{2}{G_{5}^{2}}\int
d^{5}x\sqrt{-g}\left[\frac{1}{4}{\cal{R}}-\frac{1}{2}(\nabla\phi)^{2}-V(\phi)\right]\,,$
(1)

where $G_{5}$ is five-dimensional Newton constant and ${\cal{R}}$ is the Ricci
scalar. $\phi$ and $V(\phi)$ are also the scalar field and its potential,
respectively. The non-trivial potential is chosen to be negative and has the
following form

$L^{2}V(\phi)=-3-\frac{3}{2}{\phi}^{2}-\frac{1}{3}{\phi}^{4}+\left(\frac{1}{3{\phi}_{M}^{2}}+\frac{1}{2{\phi}_{M}^{4}}\right){\phi}^{6}-\frac{1}{12{\phi}_{M}^{4}}{\phi}^{8}.$
(2)

This potential possesses a maximum at $\phi=0$ and a minimum at
$\phi={\phi}_{M}>0$, where ${\phi}_{M}$ is the model parameter. On the bulk
side each of these extrema corresponds to an AdS5 solution with different
radii. It is easily seen that the radii of these solutions take the form:

$L=\sqrt{-~{}\frac{3}{V(\phi)}}=\begin{cases}L_{UV}=L~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\phi=0~{},\\\
\\\
L_{IR}=\frac{L}{1+\frac{{\phi}_{M}^{2}}{6}}~{}~{}~{}~{}~{}\phi={\phi}_{M}~{}.\end{cases}$
(3)

From the above equation it is clearly seen that $L_{IR}<L_{UV}$. On the gauge
theory side each of these solutions are dual to the UV fixed point at $\phi=0$
and the IR fixed point at $\phi={\phi}_{M}$. According to gauge/gravity
dictionary, each of these fixed points has a number of degrees of freedom
$N^{2}$ proportional to $\frac{L^{3}}{G_{5}^{2}}$. We can calculate the
smaller number of degrees of freedom live in the IR fixed point, as indicated
by the fact that $L_{IR}<L_{UV}$. Furthermore, the difference in degrees of
freedom between the UV and IR fixed points will be increased if one increases
${\phi}_{M}$.

The vacuum solution to the Einstein equations for arbitrary ${\phi}_{M}$ can
be parameterized as:

$ds^{2}=e^{2A(r)}\left(-dt^{2}+d{\vec{x}}^{2}\right)+dr^{2}~{},$ (4)

where

$\begin{split}e^{2A(r)}=&\frac{{\Lambda}^{2}L^{2}}{{\phi}^{2}}\left(1-\frac{{\phi}^{2}}{{\phi}_{M}^{2}}\right)^{\frac{{\phi}_{M}^{2}}{6}+1}e^{\frac{-{\phi}^{2}}{6}}~{},\cr{\phi}(r)=&\frac{\Lambda
L~{}e^{\frac{-r}{L}}}{\sqrt{1+\frac{{\Lambda}^{2}L^{2}}{{\phi}_{M}^{2}}~{}e^{\frac{-2r}{L}}}}~{},\end{split}$
(5)

Where $\Lambda$ is the energy scale which breaks conformal symmetry
explicitly. By applying successive change of coordinates in the holographic
direction as Attems:2016ugt :

$\begin{split}u=&Le^{-r/L}~{},\cr
z(u)=&\int_{0}^{u}du~{}\frac{L}{u}~{}e^{-A(r)}~{},\end{split}$ (6)

we obtain the form of the metric:

$ds^{2}=\frac{L_{eff}(z)^{2}}{z^{2}}\left(-dt^{2}+d{\vec{x}}^{2}+dz^{2}\right)\,,$
(7)

where $L_{eff}=ze^{A}$ is a non-trivial function of $z$ such that
$L_{eff}(0)=L$ and $L_{eff}(\infty)=L_{IR}$.

###  II.2 Finite Temperature

Here, we review the thermal physics of the holographic non-conformal model
described in the previous section. The background solution in Eddington-
Finkelstein coordinate is Attems:2016ugt :

Figure 1: Left: Two different configurations of open string. (a): The
connected configuration: The $\cup$-shape open string configuration which
connects a quark and an anti-quark on the boundary at $z=0$ and reaches its
maximum depth in the bulk gravity at $z=z_{\ast}$. The coordinate $z$ is the
holographic direction in the bulk gravity and $z_{\ast}$ is the turning point
of the open string. (b): The disconnected configuration: The two straight open
strings that one of their endpoints is on the boundary at $z=0$ and the other
one is at the horizon $z=z_{H}$. Right: Wilson loop as the boundary of two-
dimensional string world-sheet. The blue rectangular corresponds to the Wilson
loop on the boundary of the gravity theory Yang:2015aia .

$ds^{2}=e^{2A}\left(-h(\phi)d{\tau}^{2}+d{\vec{x}}^{2}\right)-2e^{A+B}L~{}d\tau
d\phi\,,$ (8)

where $h(\phi)$ vanishes at horizon, i.e. $h(\phi_{H})=0$. Utilizing the above
ansatz and solving Einstein’s equations obtained from (1), the different
coefficients of the metric (8) are given by Attems:2016ugt :

$\begin{split}A(\phi)=&-\log\left(\frac{\phi}{\Lambda}\right)+\int_{0}^{\phi}d\tilde{\phi}\left(G(\tilde{\phi})+\frac{1}{\tilde{\phi}}\right)~{},\\\
B(\phi)=&\log(|G(\phi)|)+\int_{0}^{\phi}d\tilde{\phi}~{}\frac{2}{3G(\tilde{\phi})}~{},\\\
h(\phi)=&-\frac{e^{2B}L^{2}\left(4V(\phi)+3G(\phi)V^{\prime}(\phi)\right)}{3G^{\prime}(\phi)}~{},\end{split}$
(9)

where $G(\phi)$ is a master function defined as
$G(\phi)=\frac{d}{d\phi}A(\phi)$ and satisfies the following non-linear
equation:

$\displaystyle\frac{G^{\prime}(\phi)}{G(\phi)+\frac{4V(\phi)}{3V^{\prime}(\phi)}}=$
$\displaystyle\frac{d}{d\phi}\log\Bigg{(}\frac{1}{3G(\phi)}-2G(\phi)$
$\displaystyle+\frac{G^{\prime}(\phi)}{2G(\phi)}-\frac{G^{\prime}(\phi)}{2\left(G(\phi)+\frac{4V(\phi)}{3V^{\prime}(\phi)}\right)}\Bigg{)}\,.$
(10)

The Hawking temperature of the above solution is:

$\frac{T}{\Lambda}=-\frac{L^{2}V(\phi_{H})}{3\pi\phi_{H}}\exp\left[\int_{0}^{\phi_{H}}d\phi\left(G(\phi)+\frac{1}{\phi}+\frac{2}{3G(\phi)}\right)\right].$
(11)

##  III Wilson Loop and Heavy Quark-Antiquark Potential Energy

To do so, we utilize the Wilson loop operator. The expectation value of this
operator on a rectangular loop $\cal C$, whose two sides are $l$ and $\cal{T}$
gives us the potential energy between a static quark and an anti-quark, see
figure (1). The left panel of this figure describes the connected and
disconnected configuration of the open string and the right panel illustrates
the rectangular Wilson loop as the boundary of two-dimensional string world-
sheet. If we suppose ${\cal{T}}\gg l$ which means the world-sheet is
translationally invariant along the time direction, the expectation value of
Wilson loop is CasalderreySolana:2011us :

$\langle W({\cal{C}})\rangle=e^{-i(2m+V(l)){\cal{T}}},$ (12)

where $m$ is the rest mass of quark (or anti-quark) and $V(l)$ is the
potential energy between the pair. On the other hand, from gauge/gravity
duality we know that the expectation value of Wilson loop in the saddle point
approximation is dual with the on-shell action of classical open string that
suspended from boundary into the bulk and its endpoints correspond to a quark
and an anti-quark on the boundary separated with distance $l$:

$\langle W({\cal{C}})\rangle=e^{iS({\cal{C}})},$ (13)

where $S({\cal{C}})$ is the Nambu-Goto action:

$S_{NG}=\frac{-1}{2\pi\alpha^{\prime}}\int d\tau
d\sigma\sqrt{-\det~{}(\gamma_{s})_{ab}}~{},$ (14)

where $\alpha^{\prime}\equiv{{l}_{s}}^{2}$ in which ${l}_{s}$ is the
fundamental length of the string. In addition, we use the coordinate ($\tau$ ,
$\sigma$) to parameterize the two-dimensional world-sheet of the string and
$(\gamma_{s})_{ab}=(g_{s})_{\mu\nu}\frac{\partial
X^{\mu}}{\partial\xi^{a}}\frac{\partial X^{\nu}}{\partial\xi^{b}}$ is the
induced metric on the world-sheet. Here, $(g_{s})_{\mu\nu}$ is the bulk metric
in the string frame and $X^{\mu}$ ($\xi^{a}=\tau,\ \sigma$) are bulk (world-
sheet) coordinates, respectively.

In order to study the potential energy of the heavy quark-antiquark, it is
more convenient to use the string frame metric. One can obtain the string
frame metric via the standard method that is transformation of scalar
(dilaton) field, i.e.
$(g_{s})_{\mu\nu}=e^{\sqrt{\frac{2}{3}}\phi}(g)_{\mu\nu}$ Bohra:2019ebj ;
Critelli:2016cvq ; He:2013qq ; Gursoy:2007cb ; Gursoy:2007er . For
completeness, we mention that we considered here the scalar field $\phi$ as a
dilaton, for more details see Charmousis:2010zz  and references therein.

In order to calculate the Nambu-Goto action, $S({\cal{C}})$, we consider the
general form of the background metric as:

$ds^{2}=-f_{1}(z)dt^{2}+f_{2}(z)dz^{2}+f_{3}(z)d{\vec{x}}^{2}\,.$ (15)

we parameterize the two-dimensional string world-sheet as $\tau=t$,
$\sigma=x_{3}\equiv x$ and we choose all other bulk coordinates except $z$ and
$x$ to be constant. Therefore, $z=z(x)$ describes the shape of the string and
we set a quark and an anti-quark with distance $l$ at $x=\frac{-l}{2}$ and
$x=\frac{l}{2}$, respectively. Therefore, the action (14) reduces to:

$S_{NG}=\frac{-\cal{T}}{2\pi\alpha^{\prime}}\int_{-\frac{l}{2}}^{\frac{l}{2}}dx\sqrt{f_{1}(z)f_{3}(z)+f_{1}(z)f_{2}(z)~{}z^{\prime
2}\,}~{},$ (16)

where $z^{\prime}\\!=\\!dz/dx$. Since, the above Lagrangian does not depend
explicitly on $x$, then the corresponding Hamiltonian is a constant of motion.
Using the condition $z^{\prime}(x)|_{z=z_{\ast}}=0$ we get:

$z^{\prime}(x)=\pm\sqrt{\frac{f_{1}(z)f_{3}^{2}(z)}{f_{1}(z_{\star})f_{3}(z_{\star})f_{2}(z)}-\frac{f_{3}(z)}{f_{2}(z)}\,}~{},$
(17)

where $z_{\ast}=z(x)|_{x=0}$. Using the equation (17) we obtain:

$\int_{\pm\frac{l}{2}}^{0}~{}dx=\mp~{}\int_{\epsilon}^{z_{\star}}dz~{}\frac{1}{\sqrt{\frac{f_{1}(z)f_{3}^{2}(z)}{f_{1}(z_{\star})f_{3}(z_{\star})f_{2}(z)}-\frac{f_{3}(z)}{f_{2}(z)}\,}}~{},$
(18)

where, $\epsilon$ is the IR regulator in the gravity theory that according to
the UV/IR connection corresponds to the UV cut-off in gauge theory side
CasalderreySolana:2011us .

Figure 2:  Left: Interquark distance $l$ as a function of $z_{\ast}$ for
different values of the energy scale $\Lambda=0.30$ (Purple), $\Lambda=0.35$
(Green), $\Lambda=0.40$ (Red) and $\Lambda=0.45$ (Blue). We fix the value of
model parameter $\phi_{M}=10$. Middle: Potential energy of meson as a function
of interquark distance $l$ for different values of the energy scale
$\Lambda=0.30$ (Purple), $\Lambda=0.35$ (Green), $\Lambda=0.40$ (Red) and
$\Lambda=0.45$ (Blue). We fix the value of model parameter $\phi_{M}=10$.
Right: Relative potential energy of meson
$\tilde{V}=\frac{V-V_{AdS}}{V_{AdS}}$ as a function of $l\Lambda$ for
different values of the model parameter $\phi_{M}=5$ (blue), $\phi_{M}=7$
(Green), $\phi_{M}=10$ (red) and $\phi_{M}=20$ (Purple). We fix the interquark
distance $l=1$.

One can obtain the on-shell action by inserting equations (17) and (18) into
equation (14). In addition, the potential energy of the heavy quark-antiquark
pair can be obtained by utilizing equations (12) and (13). It is important to
note that to get a finite value for the potential energy, we need to subtract
the rest mass of the quarks from the on-shell action. The rest mass of a quark
and an anti-quark is equal to:

$2m=\frac{1}{\pi\alpha^{\prime}}\int_{\epsilon}^{zh}dz\sqrt{f_{1}(z)f_{2}(z)\,}~{}.$
(19)

Therefore, the potential energy between a quark and an anti-quark is:

$\displaystyle V=\frac{1}{\pi\alpha^{\prime}}$
$\displaystyle\Bigg{[}\int_{\epsilon}^{z_{\star}}dz\Bigg{(}\frac{f_{1}(z)\sqrt{f_{2}(z)f_{3}(z)\,}}{\sqrt{f_{1}(z)f_{3}(z)-f_{1}(z_{\star})f_{3}(z_{\star})\,}}$
$\displaystyle-\sqrt{f_{1}(z)f_{2}(z)\,}\Bigg{)}-\int_{z_{\star}}^{zh}dz\sqrt{f_{1}(z)f_{2}(z)\,}\Bigg{]}~{}.$
(20)

##  IV Numerical Results

Having set up the ingredients of studying the potential energy of the meson,
we are ready to investigate the numerical results for zero and finite
temperature cases. Hereafter, in all of our calculations we set $L=1$.

###  IV.1 Zero Temperature

At zero temperature, using the general form of the metric (15) and the metric
(7) in string frame we have:

$f_{1}(z)=f_{2}(z)=f_{3}(z)=\frac{L_{eff}(z)^{2}~{}e^{\sqrt{\frac{2}{3}}\phi}}{z^{2}}~{}.$
(21)

Therefore, one can obtain the potential energy of the heavy quark-antiquark at
zero temperature utilizing equations (III) and (21).

We plot the behavior of interquark distance $l$ as a function of $z_{\ast}$ in
the left panel of figure (2) for different values of the energy scale
$\Lambda=0.30$ (Purple), $\Lambda=0.35$ (Green), $\Lambda=0.40$ (Red) and
$\Lambda=0.45$ (Blue) and fixed value of the model parameter $\phi_{M}=10$.
According to the left panel, the corresponding potential energy of meson as a
function of interquark distance $l$ for different values of the energy scale
$\Lambda=0.30$ (Purple), $\Lambda=0.35$ (Green), $\Lambda=0.40$ (Red) and
$\Lambda=0.45$ (Blue) and fixed value of the model parameter $\phi_{M}=10$ is
depicted in the middle panel. In the right panel, we depict the relative
potential energy of meson $\tilde{V}$ as a function of $l\Lambda$ that is
defined

$\tilde{V}=\frac{V-V_{AdS}}{V_{AdS}}~{},$ (22)

where $V$ and $V_{AdS}$ are potential energy of meson corresponding to non-
conformal and AdS (conformal) background, respectively. Here, we fixed the
value of interquark distance $l=1$ and values of the model parameter
$\phi_{M}=5$ (blue), $\phi_{M}=7$ (Green), $\phi_{M}=10$ (red) and
$\phi_{M}=20$ (Purple). We list the following results.

  * •

Left panel

It is seen that by increasing $l$, a dynamical ”imaginary wall” appears in the
bulk where beyond that the connected open string world sheet does not
propagate. The physical interpretation is that, we can increase interquark
distance $l$ in such a way that the open string that connects a quark and an
anti-quark is always in the connected configuration. Accordingly, the dual
interpretation in the gauge theory side is that a quark and an anti-quark form
a bound state such that in all length scales we can investigate the Cornell
potential. In other words, we can interpret this case as the confined phase of
the theory. In addition, the location of the ”imaginary wall” shifts to higher
values of $z$ by decreasing the energy scale $\Lambda$, corresponding to a
deeper penetration into the bulk. A similar type of ”imaginary wall” is
reported before in Bohra:2019ebj . For more details see Yang:2015aia ;
Dudal:2017max ; Arefeva:2018hyo  and references therein.

  * •

Middle panel

It is seen that in the non-conformal background the meson potential energy
increases (becomes more positive) by increasing the interquark distance $l$.
We find that each of the curves in this panel can be fitted with the Cornell
potential Eichten:1978tg ; Andreev:2006ct ; Bohra:2019ebj ; Bruni:2018dqm ;
Yang:2015aia :

$V(l)=-\frac{\kappa}{l}+\sigma_{s}l+C~{},$ (23)

where, $\kappa$ is a Coulomb strength parameter, $\sigma_{s}$ is QCD string
tension and $C$ is a constant. It is clearly seen from (23) that the Coulomb
part of the Cornell potential dominates in the ultraviolet (UV) region and its
linear part dominates in the infrared (IR) region. To put in another way, the
Coulomb term, i.e. $-\frac{\kappa}{l}$ dominates for small values of $l$ and
the linear term, i.e. $\sigma_{s}l$ dominates for larger values of $l$.
Another point is that, when we increase the interquark distance $l$ we observe
that there is a specific length, i.e. $l_{\star}$, where beyond this length
the potential energy of meson experiences the linear regime of the Cornell
potential, i.e. confinement. It is seen that $l_{\star}$ increases by
increasing the energy scale $\Lambda$. Physical description of this behavior
is that when we break the conformal symmetry at larger energy scale $\Lambda$
the meson enjoys the linear confinement at larger interquark distance $l$.
Another feature is that in this panel at very small $l$ the two curves
coincide. Since, at very small $l$ we work in the UV regime where the
conformal symmetry restores and effect of the energy scale $\Lambda$ (that
breaks the conformal symmetry) disappears. Furthermore, this panel shows that
for small interquark distance $l$ (but not very small) where the potential
energy is negative, by increasing the energy scale $\Lambda$ the potential
energy between the pair decreases (becomes more negative) and the pair gets
more bounded. Interestingly, at very large interquark distance $l$ where the
linear regime of the Cornell potential is dominated, we also observe that by
increasing $\Lambda$ the potential energy increases, i.e. the interaction
between the heavy quark and antiquark increases.

  * •

Right panel

It is seen that the value of $\tilde{V}$ is always positive ($V>V_{AdS}$) and
by increasing $\Lambda$ the value of $\tilde{V}$ increases monotonically. In
addition, we observed that when the energy scale $\Lambda$ is small enough,
the relative potential energy of meson $\tilde{V}$ are approximately
independent of $\phi_{M}$. This is due to the fact that at small $\Lambda$ the
field theory is almost conformal and hence the two curves coincide. On the
other hand, at large enough $\Lambda$ the value of $\tilde{V}$ depends on
$\phi_{M}$. By increasing $\phi_{M}$, increasing the difference in degrees of
freedom between the UV and the IR fixed points, $\tilde{V}$ decreases and two
curves deviate from each other. Therefore, for small enough $\Lambda$ we can
deduce that $\tilde{V}$ is not a good measure to quantify the deviation from
the non-conformality while for large enough one we can use $\tilde{V}$ as a
measure of the non-conformality of the theory at zero temperature.

Figure 3:  Left: Interquark distance $l$ as a function of $z_{\ast}$ for
different values of the model parameter $\phi_{M}=5$ (blue), $\phi_{M}=7$
(green), $\phi_{M}=10$ (red) and $\phi_{M}=20$ (purple). We fix the value of
energy scale $\Lambda=0.4$. Right: Potential energy of meson as a function of
interquark distance $l$ for different values of given model parameter
$\phi_{M}=5$ (blue), $\phi_{M}=7$ (green), $\phi_{M}=10$ (red) and
$\phi_{M}=20$ (purple). We fix the value of energy scale $\Lambda=0.4$.

The dependence of interquark distance $l$ to the $z_{\ast}$ in the left panel
of figure (3) for fixed value of energy scale $\Lambda=0.4$ and different
values of given model parameter $\phi_{M}=5$ (blue), $\phi_{M}=7$ (green),
$\phi_{M}=10$ (red) and $\phi_{M}=20$ (purple). The corresponding meson
potential energy $V$ related to the left panel is shown as a function of
interquark distance $l$ for fixed value of energy scale $\Lambda=0.4$ and
different values of given model parameter $\phi_{M}=5$ (blue), $\phi_{M}=7$
(green), $\phi_{M}=10$ (red) and $\phi_{M}=20$ (purple). We summarize our
results in the following.

  * •

Left panel

It is seen that by increasing $l$, a dynamical ”imaginary wall” appears in the
bulk and the physical interpretation is the same as the left panel of figure
(2). In fact, beyond the ”imaginary wall” the connected open string world
sheet does not propagate. In addition, while we increase the interquark
distance $l$ the open string keeps its connected configuration and in the
gauge theory side it means a quark-antiquark pair does not dissociate and
preserves the bound state. Here, we observe this phenomena for different
values of $\phi_{M}$. In addition, in comparison to the left panel of figure
(2) the location of the ”imaginary wall” shifts to higher values of $z$ by
increasing the model parameter $\phi_{M}$, corresponding to a deeper
penetration into the bulk.

  * •

Right panel

The effect of the model parameter $\phi_{M}$ on the meson potential energy is
investigated. We observe that at fixed $\Lambda$, changing $\phi_{M}$ leads to
the change in the value of the potential energy of meson. In addition, this
panel shows that $l_{\star}$ changes by changing $\phi_{M}$. It is seen that
for $\Lambda=0.4$ for some values of $\phi_{M}$ by increasing its value the
potential energy of meson experiences the linear regime at smaller $l$.
However, this result is not general and our numerical results show that for
different values of $\Lambda$ and $\phi_{M}$ we observe different behaviors.
The interesting point is that changing $\phi_{M}$ does not change the QCD
string tension, since all the curves shown present the same slope that is in
compatible with the middle panel of the figure (4).

Figure 4: Left: Coulomb strength parameter $\kappa$ as a function of energy
scale $\Lambda$. Middle: QCD string tension $\sigma_{s}$ as a function of the
energy scale $\Lambda$. Right: The constant of the Cornell potential $C$ as a
function of energy scale $\Lambda$. Each panel is plotted for different values
of the model parameter $\phi_{M}=5$ (red), $\phi_{M}=10$ (green) and
$\phi_{M}=50$ (blue).

As we said in the previous section, the energy scale $\Lambda$ breaks the
conformal symmetry and the model parameter $\phi_{M}$ affects the difference
in degrees of freedom between the UV and IR fixed points. Next, we would like
to study the $\Lambda$ and $\phi_{M}$ dependence of Cornell potential
parameters $\kappa$, $\sigma_{s}$ and $C$ which are depicted in the left,
middle and right panels of the figure (4), respectively. These parameters are
plotted as a function of $\Lambda$ for a different values of $\phi_{M}=5$,
$\phi_{M}=10$ and $\phi_{M}=50$. Our results are listed as follows.

  * •

Left panel

We observe that the Coulomb strength parameter $\kappa$ monotonically
increases by increasing $\Lambda$ and physically it means that the value of
$\kappa$ depends on the energy scale $\Lambda$. Another point is that at small
values of $\Lambda$ the dependence of $\kappa$ on the model parameter
$\phi_{M}$ is not manifest, while for larger values of the energy scale
$\Lambda$ we see that the $\kappa$ depends on the $\phi_{M}$ clearly. Since,
$\phi_{M}$ changes the difference between the degrees of freedom between UV
and IR fixed points, i.e. $\Delta N$, we can say that the Coulomb strength
parameter $\kappa$ depends on the $\Lambda$ and $\Delta N$ that is
$\kappa(\Lambda,\Delta N)$.

  * •

Middle panel

To describe our result in this panel let’s consider the linear regime of the
Cornell potential (23) where $V\propto\sigma_{s}l$ and the QCD string tension
$\sigma_{s}$ can be obtained via $\sigma_{s}=\frac{dV}{dl}$. We observe that
the QCD string tension $\sigma_{s}$ monotonically increases when $\Lambda$
increases. To put in another word, by attention to the linear regime of the
Cornell potential we see that increasing $\Lambda$, increases the potential
energy of the pair, i.e. the pair bound together more stronger. Therefore, We
clearly observe that increasing $\Lambda$ leads to the more stable bound
state. Another important feature is that by changing $\phi_{M}$ the value of
$\sigma_{s}$ does not change that we mentioned intuitively in the physical
description of the right panel of figure (3) . Therefore, we conclude that the
$\sigma_{s}$ just depends on the energy scale $\Lambda$ that is
$\sigma_{s}(\Lambda)$.

  * •

Right panel

It is seen that the constant $C$ in the Cornell potential (23) monotonically
decreases by increasing $\Lambda$. Therefore, this means that the value of $C$
depends on the $\Lambda$. Another feature is that at small values of $\Lambda$
the dependence of $C$ on the $\phi_{M}$ is low, while for larger values of the
energy scale $\Lambda$ the $C$ depends on the $\phi_{M}$ manifestly.
Consequently, this indicates that the constant $C$ depends on the $\Lambda$
and $\Delta N$ that is $C(\Lambda,\Delta N)$.

In conclusion, in terms of the above discussion corresponding to the figure
(4), in this model the Cornell potential as a function of $l$, $\Lambda$ and
$\Delta N$ can be rewritten in the form

$V(l,\Lambda,\Delta N)=-\frac{\kappa(\Lambda,\Delta
N)}{l}+\sigma_{s}(\Lambda)l+C(\Lambda,\Delta N)~{},$ (24)

where both $\kappa$ and $C$ are function of $\Lambda$ and $\Delta N$, and
$\sigma_{s}$ is a function of $\Lambda$.

###  IV.2 Finite Temperature

At finite temperature, utilizing the metric (8) and general form of the metric
(15) in the string frame we have:

$\begin{split}f_{1}(\phi)=&h(\phi)~{}e^{2A(\phi)}e^{\sqrt{\frac{2}{3}}\phi}~{},\\\
f_{2}(\phi)=&\frac{e^{2B(\phi)}e^{\sqrt{\frac{2}{3}}\phi}}{h(\phi)}~{},\\\
f_{3}(\phi)=&e^{2A(\phi)}e^{\sqrt{\frac{2}{3}}\phi}~{}.\end{split}$ (25)

Therefore, the potential energy of meson $V$ at finite temperature $T$ can be
obtained from equations (III) and (25).

We plot $V$ as a function of $\frac{\Lambda}{T}$ for fixed values of the model
parameter $\phi_{M}=10$, interquark distance $l=0.2$ and different values of
energy scale $\Lambda=1$ (blue), $\Lambda=0.7$ (green), $\Lambda=0.5$ (red)
and $\Lambda=0.2$ (purple) in the left panel of figure (5). In the middle
panel, we plot $V$ as a function of $l{\Lambda}$ for fixed values of given
model parameter $\phi_{M}=10$, the energy scale $\Lambda=0.6$ and different
values of the temperature $T=0.0152$ (blue), $T=0.0537$ (red), $T=0.0975$
(green) and $T=0.1256$ (purple). In the right panel, we plot $V$ as a function
of $l{\Lambda}$ for different values of energy scale $\Lambda=1.2$ (blue),
$\Lambda=1$ (red), $\Lambda=0.8$ (green), $\Lambda=0.6$ (purple) and fixed
values of the given model parameter and the temperature of the system
$\phi_{M}=10$ and $T=0.0975$, respectively. Our results are listed as follows.

  * •

Left panel

It is clearly seen that the potential energy of meson as a function of
$\frac{\Lambda}{T}$ decreases monotonically. This panel shows that by
increasing the temperature $T$ the potential energy between the pair increases
and quark and antiquark can not communicate easily, i.e. the meson gets less
bounded. On the other hand, we see that by increasing $\Lambda$ (at fixed
temperature) the meson potential energy decreases and meson gets more bounded.
Therefore, this panel shows that the effect of $\Lambda$ on the potential
energy of meson $V$ is opposite to the temperature $T$. More illustration on
the low and high temperature regions is mentioned in the following.

Figure 5: Left: Potential energy of meson as a function of $\frac{\Lambda}{T}$
for different values of energy scale $\Lambda=1$ (blue), $\Lambda=0.7$
(green), $\Lambda=0.5$ (red) and $\Lambda=0.2$ (purple). We fix the values of
the given model parameter $\phi_{M}=10$ and interquark distance $l=0.2$.
Middle: Potential energy of meson as a function of $l\Lambda$ for different
values of the temperature $T=0.0152$ (blue), $T=0.0537$ (red), $T=0.0975$
(green) and $T=0.1256$ (purple). We fix the values of the given model
parameter $\phi_{M}=10$ and energy scale $\Lambda=0.6$. Right: Potential
energy of meson as a function of $l\Lambda$ for different values of energy
scale $\Lambda=1.2$ (blue), $\Lambda=1$ (red), $\Lambda=0.8$ (green) and
$\Lambda=0.6$ (purple). We fix the values of the given model parameter
$\phi_{M}=10$ and the temperature $T=0.0975$.

    * $-$

High temperature region ($\frac{\Lambda}{T}\ll 1$): The important point is
that the conformal symmetry restores and the effect of $\Lambda$ that breaks
conformal symmetry disappears. Therefore, all the curves with different
$\Lambda$ coincide. In the gravity side, the non-conformal background
approaches $AdS_{5}$.

    * $-$

Low temperature region ($\frac{\Lambda}{T}\gg 1$): The important feature is
that we can see the effect of non-conformality appears and the curves separate
from each other. This is due to the fact that in this region effect of the
energy scale $\Lambda$ dominates in comparison to the temperature T. At very
low temperature region we see that the potential energy of meson gets
approximately constant negative value. Note also that, by comparing the low
temperature region and the middle panel of figure (2) in small $l$ we conclude
that the physical effect of $\Lambda$ is the same, i.e. increasing $\Lambda$
leads to the more bounded quark-antiquark bound state.

  * •

Middle panel

It is observed that at finite temperature case the potential energy of meson
is always negative in agreement with the left panel. Another point is that, at
fixed temperature $T$ and the energy scale $\Lambda$ increasing the interquark
distance $l$ leads to the less bound state of meson. We can also observed that
the potential energy of the pair increases monotonically and reaches to the
maximum value corresponding to the special length, melting length,
$l_{m\ell}$. Beyond $l_{m\ell}$, it is clearly seen that $\cup$-shape open
string configuration of the pair in the gravity side will break into two
disconnected straight shape of open strings. Physical interpretation in the
gauge theory side is that for $l<l_{m\ell}$ we have meson bound state, while
for $l>l_{m\ell}$ the meson melting occurs in the plasma. Consequently, this
indicates that for $l>l_{m\ell}$ the potential energy is independent of $l$
suggesting $q$, $\bar{q}$ pair will be broken. It is noticed that the
connected string configuration that is appropriated for $l<l_{m\ell}$ and
describes the potential energy of bound state of a quark and an anti-quark is
denoted by a solid line, while the disconnected string configuration that is
appropriated for $l>l_{m\ell}$ is denoted by a flat dashed line. Another
feature is that, the melting length $l_{m\ell}$ is different for different
temperatures. Since, the effect of temperature $T$ and interquark distance $l$
are opposite to each other then it is anticipated that $l_{m\ell}$ occurs at
larger $l$ for lower $T$. In other words, for given values of model parameter
$\phi_{M}$ and energy scale $\Lambda$ we have
$l_{m\ell}^{T=0.0152}>l_{m\ell}^{T=0.0537}>l_{m\ell}^{T=0.0975}>l_{m\ell}^{T=0.1256}$.
In addition, when we probe the theory at very small $l$ all the curves
coincide, since the non-conformal field theory becomes conformal in the UV
limit.

  * •

Right panel

As anticipated, at small $l$ (UV limit) where we reach to the fixed point of
the theory all the curves coincide. This indicates that in this limit the
theory becomes effectively conformal. Also, in this panel similar to the
middle panel the potential energy of the quark-antiquark pair increases
monotonically and reaches to the maximum value. At intermidiate $l$ the value
of meson potential energy depends on $\Lambda$. It is clearly seen by fixing
$l$ and $T$ increasing $\Lambda$ decreases the potential energy and the meson
gets more bounded that is in agreement with the left panel. In addition,
$l_{m\ell}$ changes by varying $\Lambda$ and when we increase $\Lambda$ we see
that $l_{m\ell}$ occurs at larger values of interquark distance $l$ and lower
potential energy. In other words, for given $\phi_{M}$ and $T$ we have
$l_{m\ell}^{\Lambda=1.2}>l_{m\ell}^{\Lambda=1}>l_{m\ell}^{\Lambda=0.8}>l_{m\ell}^{\Lambda=0.6}$.
It can be described with the point that when we increase $\Lambda$, the meson
gets more stable bound state as mentioned above. Therefore, it is reasonable
that when we increase $\Lambda$, the meson dissociates at larger values of
interquark distance $l$. Note also that, from the middle and right panels we
see that the effect of $T$ and $\Lambda$ on $l_{m\ell}$ are opposite to each
other.

Figure 6:  Relative potential energy of meson
$\hat{V}=\frac{V-V_{AdS}(T=0.0438)}{V_{AdS}(T=0.0438)}$ as a function of
$\frac{\Lambda}{T}$ for different values of the model parameter $\phi_{M}=50$
(purple), $\phi_{M}=10$ (red), $\phi_{M}=7$ (green) and $\phi_{M}=5$ (blue).
We fix the values of the energy scale $\Lambda=0.4$ and interquark distance
$l=0.35$.

In figure (6), we depict the relative potential energy of meson
$\hat{V}=\frac{V-V_{AdS}(T=0.0438)}{V_{AdS}(T=0.0438)}$ at finite temperature
case for fixed values of the energy scale $\Lambda=0.4$ and interquark
distance $l=0.35$ and different values of the given model parameter
$\phi_{M}=50$ (purple), $\phi_{M}=10$ (red), $\phi_{M}=7$ (green) and
$\phi_{M}=5$ (blue). It is seen that the value of $\hat{V}$ increases
monotonically by increasing the temperature. Also, when we are in the high
temperature region ($\frac{\Lambda}{T}\ll 1$), the relative potential energy
of meson $\hat{V}$ are equal for different values of $\phi_{M}=7$ and
$\phi_{M}=20$. This is due to the fact that at high temperature the field
theory is almost conformal and hence two curves coincide. On the other hand,
at low temperature region ($\frac{\Lambda}{T}\gg 1$) the value of $\hat{V}$
depends on $\phi_{M}$. When we increase $\phi_{M}$, the value of $\hat{V}$
decreases and two curves deviate from each other. Consequently, the meson
potential energy can be considered as a measure of non-conformality of the
theory at finite temperature case.

##  V conclusion and remarks

In this paper we are interested in understanding in this research is to know
the effect of energy scale $\Lambda$, that breaks the conformal symmetry,
model parameter $\phi_{M}$, that controls the difference in degrees of freedom
between UV and IR fixed points and interquark distance $l$ on the meson
potential energy $V$ at both zero and finite temperature $T$.

Our main findings can be summarized as follows:

  * •

Interestingly, we find that parameters of the Cornell potential, i.e. $\kappa$
and $C$ depends on the $\Lambda$ and $\Delta N$ while $\sigma_{s}$ just
depends on the $\Lambda$. Therefore, in our model the Cornell potential as a
function of $l$, $\Lambda$ and $\Delta N$ can be rewritten in the form of the
equation (24).

  * •

The potential energy of meson increases by raising the interquark distance at
zero and finite temperature. At zero temperature case we observe that there is
a specific length $l_{\star}$ that beyond $l_{\star}$ the meson enjoys the
linear regime of the Cornell potential. For small values of $l$ when we
increase $\Lambda$, the meson potential energy decreases and the meson gets
more bounded while at large $l$ this behavior reverses.

  * •

By concentrating on the linear regime of the meson potential energy we find
that making larger $\Lambda$, both $l_{\star}$ and QCD string tension
$\sigma_{s}$ increase monotonically. Therefore, at larger $\Lambda$ the meson
experiences the linear regime at larger $l$.

  * •

By raising the temperature of the system the meson potential energy increases.
Also, at fixed $T$ when we increase $\Lambda$ the meson potential decreases
that is the meson gets more bounded. Therefore, the effect of $T$ and
$\Lambda$ are opposite to each other.

  * •

By investigating the meson potential energy as a function of
$\frac{\Lambda}{T}$ we observe that at very high temperature region
($\frac{\Lambda}{T}\ll 1$) all curves corresponding with different values of
$\Lambda$ or $\phi_{M}$ coincide because of restoring conformal symmetry.
Although, at very low temperature region ($\frac{\Lambda}{T}\gg 1$) all curves
deviate from each other. Also, at fixed low temperature region when we
increase $\phi_{M}$ the meson potential energy increases that is the meson
gets more bounded.

  * •

We observe that at finite temperature there is a melting length $l_{m\ell}$
where beyond $l_{m\ell}$ the meson bound state dissociates in the plasma.
Another point is that $l_{m\ell}$ is different for different temperatures of
the system, i.e. $l_{m\ell}^{T=lower}>l_{m\ell}^{T=higher}$ and occurs at
different potentials. Although, for different $\Lambda$ we have
$l_{m\ell}^{\Lambda=larger}>l_{m\ell}^{\Lambda=smaller}$. Also, our results
show that for smaller $\Lambda$ the value of $l_{m\ell}$ occurs at higher
potential.

By noting the numerical results for meson potential energy at zero and finite
temperature cases the connection between them can be considered. A very
interesting observation is that from the right panel of figure (2) and figure
(6), i.e. at both zero and finite temperature cases, the relative meson
potential energy can be considered as a good measure of non-conformality of
the theory. Furthermore, at both cases, i.e. from the middle panel of figure
(2) and the right panel of figure (5) it is observed that when we consider
small $l$, increasing $\Lambda$ decreases meson potential energy that is the
meson state gets more bounded. Also, at both cases working at very small $l$
(UV limit) all the curves coincide. This is due to the fact that conformal
symmetry restores at both cases at very small $l$. In addition, from the
middle panel of figure (2) and the left panel of figure (5) we observe that at
very low temperature and small $l$ by increasing the $\Lambda$ meson potential
energy decreases and the meson gets more bounded. Totally, our numerical
results show that at finite temperature case for any choice of parameters
increasing $\Lambda$ leads to the more stable bound state of meson while at
zero temperature case just for small and large $l$ this effect can be
observed.

It is important to note that the study of heavy quark-antiquark potential
energy by trying to reconstruct the metric via solving the equation of motion
and to fit the Cornell potential in a precise way is considered in different
models. In Pirner:2009gr  by introducing the parameter Lambda in the warp
factor with some logarithmic correction they try to reconstruct the metric and
fix the parameter Lambda in such a way that leads to the good agreement with
the Coulomb part and Linear part of the Cornell potential. In addition,
utilizing this modified metric the heavy quark-antiquark potential is
calculated for all length scales numerically. Note also that in He:2010ye  the
holographic QCD model is constructed such that the deformed warp factor is
introduced in three cases. In the case with only logarithmic correction that
the parameter Lambda is introduced in the warp factor, the heavy quark-
antiquark potential can be fitted perfectly. In the case with both logarithmic
and quadratic corrections, the Coulomb part is in good agreement with the
Cornell potential, the linear part is parallel to the Cornell potential,
however, the value of the quark-antiquark potential is a little bit higher
than the experimental data. In the case with only quadratic correction, the
resulted heavy quark-antiquark potential has both Coulomb part and linear part
and fits the Cornell potential qualitatively well. In fact, the model with
only logarithmic correction in the deformed warp factor can fit the heavy
quark potential better than the model with only quadratic correction. However,
the results of He:2010ye  show that in the case with only quadratic
correction, the Andreev-Zakharov model Andreev:2006ct  is favored to fit the
heavy quark-antiquark potential. For more details see Andreev:2006ct ;
Pirner:2009gr ; He:2010ye ; Li:2011hp ; Cai:2012xh  and references therein.

We would like to emphasize that, in our model we try to find out the
parameters of the Cornell potential numerically. In fact, we produced the
heavy quark-antiquark potential at zero and finite temperature cases for
different parameters of the theory qualitatively. In addition, we investigate
the effect of the $\Lambda$ and $\phi_{M}$ on the meson potential and the
dependence of parameters of the Cornell potential, i.e. $\kappa$, $\sigma_{s}$
and $C$ to the $\Lambda$ and $\Delta N$ at zero temperature case. Also, the
effect of the $\Lambda$, $\phi_{M}$ and $T$ on the meson potential is studied
at finite temperature case. Our approach in this research is completely
numeric and our results are qualitative.

## Acknowledgement

We would like to thank kindly for helpful comments and discussions with M.
Ali-Akbari, M. Lezgi and M. Rahimi.

## References

  * (1) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113-1133 (1999) [arXiv:hep-th/9711200 [hep-th]].
  * (2) E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253-291 (1998) [arXiv:hep-th/9802150 [hep-th]].
  * (3) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105-114 (1998) [arXiv:hep-th/9802109 [hep-th]].
  * (4) J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U. A. Wiedemann, “Gauge/String Duality, Hot QCD and Heavy Ion Collisions,” [arXiv:1101.0618 [hep-th]].
  * (5) M. Ammon and J. Erdmenger, “Gauge/gravity duality: Foundations and applications,” Cambridge Univ. Pr., Cambridge, UK, 2015.
  * (6) M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters, “Meson spectroscopy in AdS / CFT with flavor,” JHEP 07, 049 (2003) [arXiv:hep-th/0304032 [hep-th]].
  * (7) M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters, “Towards a holographic dual of large N(c) QCD,” JHEP 05, 041 (2004) [arXiv:hep-th/0311270 [hep-th]].
  * (8) S. Kobayashi, D. Mateos, S. Matsuura, R. C. Myers and R. M. Thomson, “Holographic phase transitions at finite baryon density,” JHEP 02, 016 (2007) [arXiv:hep-th/0611099 [hep-th]].
  * (9) J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik and I. Kirsch, “Chiral symmetry breaking and pions in nonsupersymmetric gauge / gravity duals,” Phys. Rev. D 69, 066007 (2004) [arXiv:hep-th/0306018 [hep-th]].
  * (10) T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog. Theor. Phys. 113, 843-882 (2005) [arXiv:hep-th/0412141 [hep-th]].
  * (11) T. Sakai and S. Sugimoto, “More on a holographic dual of QCD,” Prog. Theor. Phys. 114, 1083-1118 (2005) [arXiv:hep-th/0507073 [hep-th]].
  * (12) J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, “QCD and a holographic model of hadrons,” Phys. Rev. Lett. 95, 261602 (2005) [arXiv:hep-ph/0501128 [hep-ph]].
  * (13) A. Karch, E. Katz, D. T. Son and M. A. Stephanov, “Linear confinement and AdS/QCD,” Phys. Rev. D 74, 015005 (2006) [arXiv:hep-ph/0602229 [hep-ph]].
  * (14) D. Li, S. He, M. Huang and Q. S. Yan, “Thermodynamics of deformed AdS5 model with a positive/negative quadratic correction in graviton-dilaton system,” JHEP 09, 041 (2011) [arXiv:1103.5389 [hep-th]].
  * (15) O. DeWolfe, S. S. Gubser and C. Rosen, “A holographic critical point,” Phys. Rev. D 83, 086005 (2011) [arXiv:1012.1864 [hep-th]].
  * (16) O. DeWolfe, S. S. Gubser and C. Rosen, “Dynamic critical phenomena at a holographic critical point,” Phys. Rev. D 84, 126014 (2011) [arXiv:1108.2029 [hep-th]].
  * (17) A. Hajilou, “Meson Excitation Time as a Probe of Holographic Critical Point,” [arXiv:2111.09010 [hep-th]].
  * (18) S. Amiri-Sharifi, M. Ali-Akbari, A. Kishani-Farahani and N. Shafie, “Double Relaxation via AdS/CFT,” Nucl. Phys. B 909, 778-795 (2016) [arXiv:1601.04281 [hep-th]].
  * (19) M. Ali-Akbari, F. Charmchi, A. Davody, H. Ebrahim and L. Shahkarami, “Time-dependent meson melting in an external magnetic field,” Phys. Rev. D 91, 106008 (2015) [arXiv:1503.04439 [hep-th]].
  * (20) H. Bohra, D. Dudal, A. Hajilou and S. Mahapatra, “Chiral transition in the probe approximation from an Einstein-Maxwell-dilaton gravity model,” Phys. Rev. D 103, no.8, 086021 (2021) [arXiv:2010.04578 [hep-th]].
  * (21) D. Dudal, A. Hajilou and S. Mahapatra, “A quenched 2-flavour Einstein–Maxwell–Dilaton gauge-gravity model,” Eur. Phys. J. A 57, no.4, 142 (2021) [arXiv:2103.01185 [hep-th]].
  * (22) M. Ali-Akbari and H. Ebrahim, “Thermalization in External Magnetic Field,” JHEP 03, 045 (2013) [arXiv:1211.1637 [hep-th]].
  * (23) M. Ali-Akbari and H. Ebrahim, “Chiral symmetry breaking: To probe anisotropy and magnetic field in quark-gluon plasma,” Phys. Rev. D 89, no.6, 065029 (2014) [arXiv:1309.4715 [hep-th]].
  * (24) Z. Fang, S. He and D. Li, “Chiral and Deconfining Phase Transitions from Holographic QCD Study,” Nucl. Phys. B 907, 187-207 (2016) [arXiv:1512.04062 [hep-ph]].
  * (25) N. Callebaut, D. Dudal and H. Verschelde, “Holographic rho mesons in an external magnetic field,” JHEP 03, 033 (2013) [arXiv:1105.2217 [hep-th]].
  * (26) I. Y. Aref’eva, K. Rannu and P. Slepov, “Holographic model for heavy quarks in anisotropic hot dense QGP with external magnetic field,” JHEP 07, 161 (2021) [arXiv:2011.07023 [hep-th]].
  * (27) N. R. F. Braga, L. F. Ferreira and A. Vega, “Holographic model for charmonium dissociation,” Phys. Lett. B 774, 476-481 (2017) [arXiv:1709.05326 [hep-ph]].
  * (28) R. Abt, J. Erdmenger, N. Evans and K. S. Rigatos, “Light composite fermions from holography,” JHEP 11, 160 (2019) [arXiv:1907.09489 [hep-th]].
  * (29) T. Nakas and K. S. Rigatos, “Fermions and baryons as open-string states from brane junctions,” JHEP 12, 157 (2020) [arXiv:2010.00025 [hep-th]].
  * (30) E. Shuryak, “Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid?,” Prog. Part. Nucl. Phys. 53, 273-303 (2004) [arXiv:hep-ph/0312227 [hep-ph]].
  * (31) E. V. Shuryak, “What RHIC experiments and theory tell us about properties of quark-gluon plasma?,” Nucl. Phys. A 750, 64-83 (2005) [arXiv:hep-ph/0405066 [hep-ph]].
  * (32) M. D. Schwartz, “Quantum Field Theory and the Standard Model,”
  * (33) K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10, 2445-2459 (1974)
  * (34) J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80, 4859-4862 (1998) [arXiv:hep-th/9803002 [hep-th]].
  * (35) A. Brandhuber, N. Itzhaki, J. Sonnenschein and S. Yankielowicz, “Wilson loops in the large N limit at finite temperature,” Phys. Lett. B 434, 36-40 (1998) [arXiv:hep-th/9803137 [hep-th]].
  * (36) O. Andreev and V. I. Zakharov, “Heavy-quark potentials and AdS/QCD,” Phys. Rev. D 74, 025023 (2006) [arXiv:hep-ph/0604204 [hep-ph]].
  * (37) Y. Yang and P. H. Yuan, “Confinement-deconfinement phase transition for heavy quarks in a soft wall holographic QCD model,” JHEP 12, 161 (2015) [arXiv:1506.05930 [hep-th]].
  * (38) M. Ali-Akbari, F. Charmchi, A. Davody, H. Ebrahim and L. Shahkarami, “Evolution of Wilson loop in time-dependent N=4 super Yang-Mills plasma,” Phys. Rev. D 93, no.8, 086005 (2016) [arXiv:1510.00212 [hep-th]].
  * (39) A. Hajilou, M. Ali-Akbari and F. Charmchi, “A Classical String in Lifshitz–Vaidya Geometry,” Eur. Phys. J. C 78, no.5, 424 (2018) [arXiv:1707.00967 [hep-th]].
  * (40) T. Ishii, S. Kinoshita, K. Murata and N. Tanahashi, “Dynamical Meson Melting in Holography,” JHEP 04, 099 (2014) [arXiv:1401.5106 [hep-th]].
  * (41) A. Hajilou and M. Ali-Akbari, “Meson Excitation at Finite Chemical Potential,” Eur. Phys. J. C 79, no.3, 254 (2019) [arXiv:1804.07965 [hep-th]].
  * (42) A. Buchel, M. P. Heller and R. C. Myers, “Equilibration rates in a strongly coupled nonconformal quark-gluon plasma,” Phys. Rev. Lett. 114, no.25, 251601 (2015) [arXiv:1503.07114 [hep-th]].
  * (43) M. Lezgi, M. Ali-Akbari and M. Asadi, “Nonconformality, subregion complexity, and meson binding,” Phys. Rev. D 104, no.2, 026001 (2021) [arXiv:2011.11625 [hep-th]].
  * (44) R. A. Janik, G. Plewa, H. Soltanpanahi and M. Spalinski, “Linearized nonequilibrium dynamics in nonconformal plasma,” Phys. Rev. D 91, no.12, 126013 (2015) [arXiv:1503.07149 [hep-th]].
  * (45) U. Gursoy, M. Jarvinen and G. Policastro, “Late time behavior of non-conformal plasmas,” JHEP 01, 134 (2016) [arXiv:1507.08628 [hep-th]].
  * (46) P. Benincasa, A. Buchel and A. O. Starinets, “Sound waves in strongly coupled non-conformal gauge theory plasma,” Nucl. Phys. B 733, 160-187 (2006) [arXiv:hep-th/0507026 [hep-th]].
  * (47) M. Attems, J. Casalderrey-Solana, D. Mateos, I. Papadimitriou, D. Santos-Olivan, C. F. Sopuerta, M. Triana and M. Zilhao, “Thermodynamics, transport and relaxation in non-conformal theories,” JHEP 10, 155 (2016) [arXiv:1603.01254 [hep-th]].
  * (48) M. Rahimi, M. Ali-Akbari and M. Lezgi, “Entanglement entropy in a non-conformal background,” Phys. Lett. B 771, 583-587 (2017) [arXiv:1610.01835 [hep-th]].
  * (49) M. Ali-Akbari, M. Rahimi and M. Asadi, “Holographic mutual and tripartite information in a non-conformal background,” Nucl. Phys. B 964, 115329 (2021) [arXiv:1907.08917 [hep-th]].
  * (50) M. Asadi, “On volume subregion complexity in non-conformal theories,” Eur. Phys. J. C 80, no.7, 681 (2020) [arXiv:2004.11306 [hep-th]].
  * (51) R. Critelli, R. Rougemont, S. I. Finazzo and J. Noronha, “Polyakov loop and heavy quark entropy in strong magnetic fields from holographic black hole engineering,” Phys. Rev. D 94, no.12, 125019 (2016) [arXiv:1606.09484 [hep-ph]].
  * (52) H. Bohra, D. Dudal, A. Hajilou and S. Mahapatra, “Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model,” Phys. Lett. B 801, 135184 (2020) [arXiv:1907.01852 [hep-th]].
  * (53) S. He, S. Y. Wu, Y. Yang and P. H. Yuan, “Phase Structure in a Dynamical Soft-Wall Holographic QCD Model,” JHEP 04, 093 (2013) [arXiv:1301.0385 [hep-th]].
  * (54) U. Gursoy and E. Kiritsis, “Exploring improved holographic theories for QCD: Part I,” JHEP 02, 032 (2008) [arXiv:0707.1324 [hep-th]].
  * (55) U. Gursoy, E. Kiritsis and F. Nitti, “Exploring improved holographic theories for QCD: Part II,” JHEP 02, 019 (2008) [arXiv:0707.1349 [hep-th]].
  * (56) C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, “Effective Holographic Theories for low-temperature condensed matter systems,” JHEP 11, 151 (2010) [arXiv:1005.4690 [hep-th]].
  * (57) D. Dudal and S. Mahapatra, “Thermal entropy of a quark-antiquark pair above and below deconfinement from a dynamical holographic QCD model,” Phys. Rev. D 96, no.12, 126010 (2017) [arXiv:1708.06995 [hep-th]].
  * (58) I. Aref’eva and K. Rannu, “Holographic Anisotropic Background with Confinement-Deconfinement Phase Transition,” JHEP 05, 206 (2018) [arXiv:1802.05652 [hep-th]].
  * (59) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, “Charmonium: The Model,” Phys. Rev. D 17, 3090 (1978).
  * (60) R. C. L. Bruni, E. Folco Capossoli and H. Boschi-Filho, “Quark-antiquark potential from a deformed AdS/QCD,” Adv. High Energy Phys. 2019, 1901659 (2019) [arXiv:1806.05720 [hep-th]].
  * (61) H. J. Pirner and B. Galow, “Strong Equivalence of the AdS-Metric and the QCD Running Coupling,” Phys. Lett. B 679, 51-55 (2009) [arXiv:0903.2701 [hep-ph]].
  * (62) S. He, M. Huang and Q. S. Yan, “Logarithmic correction in the deformed $\rm AdS_{5}$ model to produce the heavy quark potential and QCD beta function,” Phys. Rev. D 83, 045034 (2011) [arXiv:1004.1880 [hep-ph]].
  * (63) D. Li, S. He, M. Huang and Q. S. Yan, “Thermodynamics of deformed AdS5 model with a positive/negative quadratic correction in graviton-dilaton system,” JHEP 09, 041 (2011) [arXiv:1103.5389 [hep-th]].
  * (64) R. G. Cai, S. He and D. Li, “A hQCD model and its phase diagram in Einstein-Maxwell-Dilaton system,” JHEP 03, 033 (2012) [arXiv:1201.0820 [hep-th]].