The endomorphism ring of a square-free  injective module]The endomorphism ring of a square-free
injective module

Dedicated to Professor Alberto Facchini for his 60th birthday
Mai Hoang Bien
Mathematisch Instituut, Leiden Universiteit, Niels Bohrweg 1,2333 CA Leiden,The Netherlands.
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy.

Let $M_R$ be an injective right module over a ring $R$. The goal of this paper to prove that the endomorphism ring $S=\End(M_R)$ of $M_R$ is quasi-duo if and only if $M_R$ is square-free.

§ INTRODUCTION

In this paper, rings are meant to be associative with identity and modules are understood to be right modules. It is well known that an injective module is indecomposable if and only if its endomorphism ring is a local ring [4]. The aim of this paper is to generalize this result from indecomposable modules to square-free ones.

Recall that an injective $R$-module $M_R$ is said to be square-free if it contains no direct summand isomorphic to $X\+X$ for some direct summand $0\ne X$ of $M_R$. A ring $S$ is called left quasi-duo (resp. right quasi-duo) if every maximal left ideal (resp. maximal right ideal) of $S$ is a two-sided ideal.  If $S$ is both left and right quasi-duo then $S$ is said to be quasi-duo. The class of quasi-duo rings is not “small". For example, commutative rings and local rings are quasi-duo. Factors of a quasi-duo ring are also quasi-duo. In [5], the author proved that $S$ is quasi-duo if and only if $S/J(S)$ is quasi-duo. Here, $J(S)$ is the Jacobson radical of $S$. The direct product of a family of rings is quasi-duo if and only if each ring is quasi-duo. The matrices ring $M_n(D)$ of degree $n$ over a division ring $D$ is quasi-duo if and only if $n=1$. Hence, a direct product of finitely many division rings is quasi-duo. Therefore, rings of finite type are quasi-duo [1]. In particular, the endomorphism rings of univerial modules, couniform projective modules, cyclically presented modules over a local ring, and the kernels of morphisms between indecomposable injective modules  are quasi-duo (see [1]). Examples of right quasi-duo rings which are not left quasi-duo are unknown <cit.>. Of course, if there exists such ring $R$ then the opositive ring $R^{op}$ of $R$ is left quasi-duo and not right quasi-duo.

Let $M_R$ be an injective module over $R$. In this paper, we will show that $M_R$ is square-free if the endomorphism ring $S: =\End(M_R)$ of $M_R$ is either left quasi-duo or right quasi-duo (Theorem <ref>). Conversely, if $M_R$ is square-free then $S$ is quasi-duo. In fact, we prove that if $M_R$ is square-free then every one-sided ideal of $S$ containing $J(S)$ is two-sided (Theorem <ref> and <ref>).

The symbols and notations we use in this paper are familiar to module and ring theory. $\subseteq _e $ will denote an essential submodule, and $E(N)$ will denote an injective envelope of a module $N$. For any direct summand $N$ of a $R$-module $M_R$, $\iota_N$ and $\pi_N$, respectively, are the embedding of $N$ into $M_R$ and the projection of $M_R$ onto $N$.

§ RESULTS

Let $M_R$ be an injective $R$-module. If the endomorphism ring $S=\End(M_R)$ of $M_R$ is either left quasi-duo or right quasi-duo then $M_R$ is square-free.

Assume that $S$ is left quasi-duo and there exist direct summands $A,B,C$ of $M_R$ such that $A\+B\+C=M_R$ and $A$ is isomorphic to $B$. Call $I$ a maximal left ideal of $S$ containing $s=\iota_{B\+C}\pi_{B\+C}$. Then $I$ is a two-sided ideal of $S$ by hypothesis. Let $\alpha$ be an isomorphism from $A$ to $B$. Define $f:=\iota_B\alpha\pi_A$ and $g:=\iota_A\alpha^{-1}\pi_B+\iota_C\pi_C$. Then $s f+g s\in I$ and

$$\ker (s f+gs) =\{\, a+b+c\in A\+B\+C\mid (s f+gs)(a+b+c)=0 \,\}$$

$$=\{\, a+b+c\in A\+B\+C \mid \alpha (a)+\alpha^{-1}(b)+c=0 \,\}=0.$$
Hence, $s f+gs$ is injective. Consider the diagram

\xymatrix{
0 \ar[r] &{A\+B\+C=M_R}\ar[r]^{sf+gs}  \ar[d]^{1} &M_R<EMAIL_ADDRESS>   \\
Since $M_R$ is injective, there exists morphism $h$ of $S$ such that $h(s f+gs)=1$, which imlies $I=S$. Contradiction. Therefore, if $S$ is left quasi-duo then $M_R$ is square-free.

Similarly to the right side.

Let $M_R$ be a  square-free injective $R$-module. If $N_1, N_2$ are two isomorphic direct summands of $M_R$ then $N_1$ and $N_2$ are two injective envelopes of $N_1\cap N_2$.

Let $A_1, A_2$ be injective envelopes of $N_1\cap N_2$ respectively in $N_1$, $N_2$ and $B_1,B_2$ be respectively direct summands of $N_1,N_2$ such that
$$N_1=A_1\+B_1, N_2=A_2\+B_2.$$
Since $N_1\cong N_2$ and $B_1\cap A_2=0$,   $B_1$ is isomorphic to a direct summand $C$ of $N_2$ and since $B_1\cap N_2=0$, $B_1+N_2=B_1\+N_2$ is injective, which implies that $M_R$ contains $B_1\+C$, with $B_1\cong C$, as a direct summand. Therefore $B_1=0$. Similarly, $B_2=0$. Thus, $N_1, N_2$ are two injective envelopes of $N_1\cap N_2$.

Let $M_R$ be a square-free injective $R$-module with $M_R=M_1\+M_2$. If $N$ is a direct summand of $M_R$ then $N$ is an injective envelope of $(N\cap M_1)\+(N\cap M_2)$.

Let $A$ be an injective envelope of $N\cap M_1$ in $N$ and $B$ be a direct summand of $N$ such that $N=A\+B$. Consider $\pi:=\pi_{M_2}$, the projection of $M_1\+M_2$ onto $ M_2$, and the morphism $\pi_{\left.   \right| B}:B\to M_2$ restricted on $B$ of $\pi$. Because $B\cap M_1=0$, ${\left. \pi  \right|_B}$ is injective. Hence, $B$ is isomorphic to a direct summand $C$ of $M_2$. By Lemma <ref>, $B$ is an injective envelope of $B\cap C$. Therefore, $B$ is an injective envelope of $N\cap M_2=B\cap M_2\supseteq B\cap C$. This implies $N$ is an injective envelope of $(N\cap M_1)\+(N\cap M_2)$.

Let $M_R$ be an injective $R$-module, $S=\End(M_R)$ be the endomorphism ring of $M_R$ and $J(S)$ be the Jacobson radical of $S$. For any element $f\in S$, there exist $e_1,e_2, g_1,g_2, h_1,h_2\in S$ and $ i_1, i_2, j_1, j_2\in J(S)$ such that $e_1,e_2$ are idempotents and
$$e_1=fg_1+i_1, f=e_1h_1+j_1,$$

$$e_2=g_2f+i_2, f=h_2e_2+j_2.$$

It is well-known that $S/J(S)$ is a Von Neumann regular ring and any idempotent of $S/J(S)$ can be lifted to an idempotent of $S$ (see <cit.>). Hence, for any $f\in S$, there exists idempotent $e_1\in S$ such that
Therefore, $e_1=fg_1+i_1, f=e_1h_1+j_1$ for some $g_1,h_1\in S$ and $i_1,j_1\in J(S)$. Similarly to $e_2=g_2f+i_2, f=h_2e_2+j_2.$

Let $M_R$ be an injective $R$-module, $S=\End(M_R)$ be the endomorphism ring of $M_R$ and $e$ be an idempotent of $S$. For any element $f\in S$, $f\in eS$ if and only if $f(M_R)\subseteq e(M_R)$.

Assume that $f=eh$ for some $h\in S$. Then $f(M_R)=eh(M_R)\subseteq e(M_R)$. Conversely, assume that $f(M_R)\subseteq e(M_R)$. Then for any $x\in M_R$, $f(x)=e(y)$ for some $y\in M_R$. Hence,
Therefore, $f=ef\in eS$.

Let $M_R$ be an injective $R$-module and $S=\End(M_R)$ be the endomorphism ring of $M_R$. If $M_R$ is square-free then every right ideal of $S$ containing $J(S)$ of $S$ is a two-sided ideal. In particular, if $M_R$ is square-free then $S$ is right quasi-duo.

Let $I$ be a right ideal of $S$ containing $J(S)$. Let $f\in I$ and $\phi\in S$. We must show that $\phi f\in I$. By Lemma <ref>, there exist $e,g,h\in S$ and $i,j\in J(S)$ such that $e=fg+i$ is an idempotent and $f=eh+j$. We have $e$ belongs to $I$ and $\phi f=\phi e h+\phi j$, so that it suffices to show that $\phi e\in I$. Indeed, let $N'$ be a direct summand of $M_R$ such that $M_R=N\+N'$ with $N=e(M_R)$. If set $M_1$ is an injective envelope of $\ker \phi$, then there exists a direct summand $M_2$ of $M_R$ such that $M_R=M_1\+M_2$.  Let $N_1, N_2$ be respectively injective envelopes of $N\cap M_1, N\cap M_2$ in $N$. By Lemma <ref>, we may assume that
Consider the morphism $\phi _{\left. \right|{N_2}}:N_2\to N_1\+N_2\+N'$ restricted on $N_2$ of $\phi$. It is easy to check that $\phi _{\left. \right|{N_2}}$ is injective.  Hence, $N_2\cong \phi _{\left. \right|{N_2}}(N_2)$ and by Lemma <ref>, $A=N_2\cap \phi _{\left. \right|{N_2}}(N_2)$ is an essential submodule of $N_2$. Set $B=\phi _{\left. \right|{A}}^{-1}(A)$ with $\left. \phi\right|_{A}:A\to N_2$ is the morphism restricted on $A$ of $\phi$. Then $B$ is also an essential submodule of $N_2$. Write $\psi: N_2\to N_2$ for a morphism extending of $\phi _{\left. \right|{A}}$ and let $\psi'=\iota_{N_2}\psi\pi_{N_2}$. One has $\psi'(M_R)\subseteq N_2\subseteq N=e(M_R)$, which implies from Lemma <ref> that $\psi'\in eS\subseteq I$. Moreover, since for any $a+b+c\in (\ker \phi\cap N_1)\+B\+N'$, $(\phi e-\psi')(a+b+c)=0$ and $(\ker \phi\cap N_1)\+B\+N'$ is essential in $M_R$, $\phi e-\psi'$ belongs to $J(S)$. This shows that $\phi e\in I$.

Let $M_R$ be an injective $R$-module, $S=\End(M_R)$ be the endomorphism ring of $M_R$. Then for any  two elements  $f,g$ of $S$,  $g=hf+j$ for some $h\in S, j\in J(R)$ if and only if $\ker f\cap A \subseteq \ker g$ for some essential submodule $A$ of $M_R$.

Assume that $g=hf+j$ for some $h\in S$ and $j\in J(S)$. Then $\ker g\supseteq \ker (hf)\cap \ker j \supseteq \ker f\cap A$ with $A=\ker(j)\subseteq_e M_R$.

Conversely, assume that $\ker f\cap A\subseteq \ker g$ for a submodule $A\subseteq _eM_R$. Let $M_1$ be an injective envelope of $\ker g$ in $M_R$, and $N_1$ be an injective envelope of $\ker f \cap A$ in $M_1$. Assume that $M_R=N_1\+N_2\+M_2$ for some direct summands $N_2, M_2$ respectively of $M_1$ and $M_R$. Consider the diagram

\xymatrix{
0 \ar[r] &{N_2\+M_2}\ar[r]^{f_{\left.  \right|{N_2 \oplus M_2}}}  \ar[d]^{g_{\left.  \right|{N_2 \oplus M_2}}} &M_R<EMAIL_ADDRESS>   \\
Here, ${f_{\left.  \right|{N_2 \oplus M_2}}}, {g_{\left.  \right|{N_2 \oplus M_2}}}$ are the morphisms respectively restricted on $N_2\+M_2$  of $f,g$. Since $M_R$ is injective, there exists $h:M_R\to M_R$ such that
$${g_{\left.  \right|{N_2 \oplus M_2}}}=h.{f_{\left.  \right|{N_2 \oplus M_2}}}.$$
Because $(g-hf)(a+b+c)=0$ for any  $a+b+c\in (\ker f\cap A)\+N_2\+M_2$ which is essential in $M_R$, $j=g-hf\in J(S)$.

Let $M_R$ be an injective $R$-module and $S=\End(M_R)$ be the endomorphism ring of $M_R$. If $M_R$ is square-free then every left ideal of $S$ containing the Jacobson radical $J(S)$ of $S$ is a two-sided ideal. In particular, if $M_R$ is square-free then $S$ is left quasi-duo.

Let $I$ be a left ideal of $S$ containing $J(S)$. Let $f,\phi$ be elements of $ S$ with $f\in I$. We must show that $f\phi \in I$. By Lemma <ref>, there exist $e,g,h\in S$ and $i,j\in J(S)$ such that $e=gf+i$ is an idempotent and $f=he+j$. Hence, $e\in I$ and $f\phi =he \phi+j\phi$. It suffices to show that $e\phi \in I$. Indeed, set $N_1=\ker e$ and let $N_2$ be a direct summand of $M_R$ such that $M_R=N_1\+N_2$. Consider $\psi:=\pi_{N_2}\phi\iota_{N_1}:N_1\to N_2$. If set $N_1'$ is an injective envelope of $A:=\ker \phi$ in $N_1$ then exists a direct summand $N_1''$ of $M_R$ such that $N_1=N_1'\+N_1''$. Since $\ker \psi_{\left.  \right|{{N_1''}}}=N_1''\cap A=0$, $\psi_{\left.  \right|{{N_1''}}}$ is injective. This implies $N_1''$ is isomorphic to a direct summand $C$ of $N_2$. Hence, $M_R$ contains a direct summand isomorphic to $C\+C$. By hypothesis, $C=0$. In other words, $A$ is essential in $N_1$. Now, one has that $\phi$ can be written as the matrix $\phi=\left( {\begin{array}{*{20}{c}}
\end{array}} \right)$. Then $\ker (e\phi)=\phi^{-1}(\ker e)=\phi^{-1}(N_1)\supseteq \ker(\pi_{N_2}\phi\iota_1)\+\ker(\pi_{N_2}\phi\iota_{N_2})\supseteq A=(A\+N_2)\cap N_1=(A\+N_2)\cap \ker e$. Since $A\+N_2$ is essential in $M_R$, by Lemma <ref>, there exist $h\in S, j\in J(S)$ such that $e\phi=he+j$. Thus $e\phi\in I$.

[1] Facchini, A.: Direct-sum decompositions of modules with semilocal endomorphism rings. Bull. Math. Sci. 2  (2), , 225–279, (2013).

[2] Lam, T.Y.: Lectures on modules and rings. Graduate Texts in Mathematics, Vol. 189, Springer, Berlin, Heidelberg, New York (1998).

[3] Lam, T.Y, Dugas, A.S.: Quasi-duo rings and stable range descent. J. Pure Appl. Algebra. 195 (3), 243–259 (2005).

[4] Matlis, E.: Injective modules over Noetherian rings. Pacific J. Math. 8, 51–528 ( 1958).

[5] Yu, H.-P.:  On quasi-duo rings. Glasgow Math. J. 1 37 (1), 21–31 (1995).


††thanks: Corresponding author

#  Configuration entropy of $\Upsilon(1S)$ state in strong coupling plasma

Yan-Qing Zhao<EMAIL_ADDRESS>Institute of Particle Physics and
Key Laboratory of Quark and Lepton Physics (MOS), Central China Normal
University, Wuhan 430079, China  Defu Hou<EMAIL_ADDRESS>Institute of
Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS), Central
China Normal University, Wuhan 430079, China

###### Abstract

To better understand the effects of strong coupling and QCD at high
temperature in QGP, by using holographic model, we investigate the
dissociation effect of bottomonium under the higher-order curvature
corrections to the supergravity action corresponding to the corrections of
large N expansion of boundary CFT in the side of field theory. The results
show that effective potential is not a good physical quantity to estimate the
dissociation strength of bottomonium in the case of finite wave number and
considering the higher-order curvature corrections. Therefore, we calculate
the quasinormal spectra(QNMs) and the differential configuration entropy(DCE).
It is found that the dissociation effect is stronger for the stronger
coupling.

Configuration Entropy, Heavy Vector Mesons, Higher Derivative Corrections,
Gauss-Bonnet Gravity

##  I  Introduction

To investigate the formation and evolution of quark gluon plasma(QGP), the
study of heavy quarkonium($J/\psi$ and $\Upsilon(1S)$ mainly) production rate
plays an irreplaceable role in the process. Because the QGP matter can not be
observed directly. It is well-known that heavy vector mesons may still survive
when light mesons have dissolved, that is to say, the physical system has
undergone the deconfinement phase transition. So far, there are many physical
quantities and methods that can be used to study the properties of heavy
flavors. For example, the real potential Finazzo and Noronha (2015); Kim et
al. (2009); Hou and Ren (2008); Boschi-Filho et al. (2006); Feng et al.
(2020), imaginary potential Zhao et al. (2020); Feng et al. (2020); Zhang and
Zhu (2019); Bitaghsir Fadafan and Tabatabaei (2016); Finazzo and Noronha
(2015), thermal width Feng et al. (2020); Braga and Ferreira (2016); Bitaghsir
Fadafan and Tabatabaei (2016); Finazzo and Noronha (2015); Fadafan and
Tabatabaei (2014); Zhao et al. (2020), separation length Feng et al. (2020);
Liu et al. (2007a, b), thermal spectral functions Braga and Ferreira (2017);
Zhao and Hou (2022) of heavy quarkonium and so on.

In recent years a great deal of interest has been shown in using the
configuration entropy(CE) to study the stability of $Q\bar{Q}$ pair, which is
put forward by Gleiser and Stamatopoulos in refs. Gleiser and Stamatopoulos
(2012a, b); Gleiser and Sowinski (2013). Configuration entropy is closely
related to quasinormal modes. In order to get the configuration entropy, one
must look for complex frequencies of the heavy quarkonium by solving the
equation of motion of quasi-states with related boundary condition. The real
part of the complex frequency are associated with the thermal mass and the
imaginary part with the thermal width of heavy quarkonium. Many research
results show that the more stable the system, the smaller the CE value Gleiser
and Jiang (2015); Bernardini and da Rocha (2016); Bernardini et al. (2017);
Braga and da Rocha (2018); Braga et al. (2018); Bernardini and Da Rocha
(2018); Ferreira and Da Rocha (2019); Alves et al. (2015); Correa and da Rocha
(2015); Gleiser and Stamatopoulos (2012a); Correa and Moraes (2016); Braga and
da Rocha (2017, 2018); Karapetyan (2017, 2018a, 2018b); Bazeia et al. (2019);
Lee (2017); Colangelo and Loparco (2019); Lee (2019); Bernardini and da Rocha
(2019); Karapetyan (2020, 2019); Gleiser and Krackow (2020); Cruz et al.
(2019a, b); Bhattacharjee (2020). In ref. Braga and da Mata (2020a),
configuration entropy of charmonium is calculated by considering the influence
of magnetic fields. In ref. Braga and da Mata (2020b), they develop the
configuration entropy for quarkonium in a finite density plasma. The
definition of configuration entropy originated from Shannon information
entropy Shannon (1948) that gives the information contained in the variable
$a$, which may have the discrete value $a_{n}$ and the probabilities $p_{n}$,

$S[f]=-\sum_{n}p_{n}\log p_{n},$ (1)

where $p_{n}$ satisfies the normalization condition: $\sum_{n}p_{n}=1$. For
successive quantity, Shannon information entropy reads

$S[f]=-\int d^{d}x\epsilon(\vec{x})\log\epsilon(\vec{x}),$ (2)

where modal fraction,

$\epsilon(\vec{x})=\frac{|{\rho(\vec{x})}|^{2}}{\int
d^{d}x|{\rho(\vec{x})}|^{2}},$ (3)

satisfies normalization condition $\int d^{d}r\epsilon(\vec{x})=1$. The
normalizable function in coordinate space $\rho(\vec{x})$ needs to be
transformed into momentum space by Fourier transformation for the sake of
calculating the configuration entropy,

$\rho(\vec{k})=\frac{1}{(2\pi)^{d/2}}\int
d^{d}x\rho(\vec{x})e^{-i\vec{k}\cdot\vec{x}}.$ (4)

Accordingly, the modal fraction is given by

$\overline{\epsilon}(\vec{k})=\frac{|{\rho(\vec{k})}|^{2}}{\int
d^{d}k|{\rho(\vec{k})}|^{2}}.$ (5)

Furthermore, the configuration entropy is introduced as follows

$S_{CE}=-\int
d^{d}k\overline{\epsilon}(\vec{k})\log\overline{\epsilon}(\vec{k}).$ (6)

However, the value of CE may be negative for continuous variables. For
convenience, another different modal fraction was introduced in Gleiser et al.
(2018)

$\epsilon(\vec{k})=\frac{|{\rho(\vec{k})}|^{2}}{|{\rho(\vec{k})}|^{2}_{max}}.$
(7)

The resulting configuration entropy is called differential configuration
entropy(DCE),

$S_{DCE}=-\int d^{d}k\epsilon(\vec{k})\log\epsilon(\vec{k}).$ (8)

The study of quark-gluon plasma (QGP) is of great importance in understanding
the fundamental properties of matter under extreme conditions. Holographic
methods have provided a powerful tool for investigating QGP from a
gravitational perspective. However, the effects of strong coupling and QCD at
high temperature in QGP are still not well understood. To address this issue,
we consider the high-order curvature corrections in the gravity theory when
studying QGP using holographic methods. By doing so, we aim to gain a deeper
insight into the behavior of QGP and uncover new and interesting phenomena,
such as strong fluid properties and perturbation theories, which may have
important implications for our understanding of QGP and the nature of matter
at extreme conditions.

The main objective of the present study is to explore the quasinormal
frequencies and the differential configuration entropy of heavy quarkonia in a
hot dense matter by considering a class of bulk theories comprising higher-
order curvature corrections to the supergravity action Cai (2002), which
results in the conjectured viscosity bound Kovtun et al. (2005)
$\frac{\eta}{s}\geq\frac{1}{4\pi}$ to be violated. In the effective low-energy
action of string theories, according to AdS/CFT dual, the higher-order
curvature term can be regarded as the corrections of large N expansion of
boundary CFT in the side of corresponding field theory. Using the holographic
model for $\Upsilon(1S)$ in a hot dense plasma, the QNMs and the DCE of
bottomonium are calculated. We will discuss the dissociation behavior of
$Q\bar{Q}$ pair and possible implications on higher-order curvature
corrections related effects.

##  II  The holographic model

For going deeply into higher-order curvature corrections, we take into
account, a most general theory of gravity with quadratic powers of curvature
in five-dimensional $(5D)$ spaces,Gauss-Bonnet(GB) gravity. The action is
Zwiebach (1985); Buchel et al. (2010)

$\displaystyle S=\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-G}[(\mathcal{R}+\frac{12}{R^{2}})+\frac{\lambda_{GB}R^{2}}{2}\mathcal{L}_{GB}],$
(9)
$\displaystyle\mathcal{L}_{GB}=\mathcal{R}^{2}-4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}.$
(10)

Here $G_{5}$ is the five dimensional Newton constant, $R$ represents the
radius of the asymptotic AdS spaces, $\mathcal{R}_{\mu\nu}$ labels the Ricci
tensor and $\mu$ runs over 1 to 5, $\mathcal{R}_{\mu\nu\rho\sigma}$ denotes
the Riemann tensor, $\mathcal{R}$ refers to the Ricci scalar, and
$\lambda_{GB}$ is a coupling constant. The first polynomial gives the
Einstein-Hilbert action and $\mathcal{L}_{GB}$ is the $\mathcal{R}^{2}$
corrections. The action exists a pinpoint black-brane solution, the metric can
be written as Cai (2002)

$ds^{2}=\frac{R^{2}}{z^{2}}(-af_{GB}(z)dt^{2}+dx^{2}+\frac{1}{f_{GB}(z)}dz^{2})$
(11)

with

$a=\frac{1}{2}(1+\sqrt{1-4\lambda_{GB}}),$ (12)
$f_{GB}(z)=\frac{1}{2\lambda_{GB}}(1-\sqrt{1-4\lambda_{GB}(1-\frac{z^{4}}{z_{h}^{4}})}).$
(13)

The black brane horizon $z_{h}$ is at $f_{GB}(z_{h})=0$. Ref. Cai (2002)
points out that in five-dimensional spaces $\lambda_{GB}\leq 9/100$, which can
validly avoid causality violation at the boundary Brigante et al. (2008a, b).
One can find that there is a vacuum AdS solution only when $\lambda_{GB}\leq
0.25$ and exists a conformal field theory at the boundary. The Hawking
temperature of the plasma can be obtained by the surface gravity, as follows

$T=\absolutevalue{\frac{\kappa}{2\pi}}=\frac{\sqrt{a}}{\pi z_{h}},$ (14)

where
$\kappa=\sqrt{-g_{tt}/g_{zz}}(\text{log}(\sqrt{-g_{tt}}))^{\prime}|_{z=z_{h}}$
is the surface gravity. In Fig.1, we show the behavior of black hole
temperature $T$ as a function of the location of horizon $z_{h}$. As can be
seen, increasing the location of horizon reduces the temperature, and the
appearance of higher-order curvature corrections also diminishes the black
hole temperature.

The constant $\lambda_{GB}$ and the ratio of the shear viscosity $\eta$ and
the entropy density $s$ has the following relation Brigante et al. (2008a, b);
Kats and Petrov (2009)

$\frac{\eta}{s}=\frac{1}{4\pi}(1-4\lambda_{GB}).$ (15)

The viscosity bound, $\eta/s>1/4\pi$, is violated for $\lambda_{GB}>0$. The
restraint $\lambda_{GB}\leq 9/100$ indicates that $\eta/s\leq 4/(25\pi)$. As
can be seen from eq. (15), for the same physical quantity, the behavior of the
ratio $\eta/s$ and the constant $\lambda_{GB}$ is opposite. For convenience,
in the following sections, we all use $\lambda_{GB}$ instead of $\eta/s$ to
participate in the calculation.

Figure 1:  The location of horizon $z_{h}$ dependence of black hole
temperature $T$ with different constants $\lambda_{GB}$. The red curve
represents $\lambda_{GB}=-0.19$; the blue curve is the result for
$\lambda_{GB}=0$ and the magenta curve denotes $\lambda_{GB}=0.09$.

Now, with the help of the phenomenological holographic model of Braga and
Ferreira (2019), we research the nature of heavy vector mesons in a hot dense
plasma. We use a 5D dual vector field $A_{m}=(A_{\mu},A_{z})(\mu=0,1,2,3)$ to
represent the heavy quarkonium. The action takes the following form

$S_{\text{matter}}=\int
d^{4}xdz\sqrt{-g}e^{-\phi(z)}[-\frac{1}{4{g_{5}}^{2}}F_{mn}F^{mn}],$ (16)

where $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$. $\phi(z)$ is a background
scalar field that includes three energy parameters,

$\phi(z)=m_{b}^{2}z^{2}+M_{b}z+\tanh(\frac{1}{M_{b}z}-\frac{m_{b}}{\sqrt{\Gamma_{b}}}).$
(17)

The three energy parameters $m_{b}$ expresses the bottom quark mass,
$\Gamma_{b}$ means the string tension of the quark pair($b\bar{b}$) and
$M_{b}$ represents a large mass related to the bottomonium non-hadronic decay.
There exists a matrix element $\langle
0|J_{\mu}(0)|\Upsilon(1S)\rangle=\epsilon_{\mu}f_{n}m_{n}$ ($\langle 0|$ is
the hadronic vacuum, $J_{\mu}$ is the hadronic current and $f_{n}$ indicates
the decay constant) when the bottomonium decays into leptons. The value of
three energy parameters for bottomonium in the background scalar field,
provided by fitting the spectrum of masses in Braga and Ferreira (2018), are
respectively:

$\displaystyle m_{b}=2.45GeV,\,\,\sqrt{\Gamma_{b}}=1.55GeV,\,\,M_{b}=6.2GeV.$
(18)

##  III  The differential configuration entropy

With all the preparations in place, let’s calculate the differential
configuration entropy for heavy vector mesons. The quantity $\rho(\vec{x})$ in
eq. (4) is the energy density which is the $T_{00}$ component of the energy-
momentum tensor. The energy tensor from action (16) can be expressed in the
following form with metric (11) :

$T_{\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{matter}}}{\delta
g^{\mu\nu}}.$ (19)

Here, we consider gauge $A_{z}=0$, and the remaining components
$A_{\mu}=A_{\mu}(p,z)e^{-i\omega t+ipx_{3}}$ which corresponds to plane waves
propagating in the $x_{3}$ direction with momentum $p$. Futher, the field
$A_{\mu}(p,z)$ can be written as $A_{\mu}(p,z)=\eta_{\mu}v(p,z)$ with the
polarization vector $\eta_{\mu}$, where $v(p,z)$ is known as the bulk-to-
boundary propagator corresponding to the quasinormal modes, satisfing
$v(p,z)|_{z\rightarrow 0}=1$. The boundary value of the vector field
$A_{\mu}(p,z)|_{z\rightarrow 0}=\eta_{\mu}$ can be seen as the source of the
correlation functions of boundary current operator. In the zero temperature
limit, the corresponding solutions are called the normal modes representing
the vacuum hadronic states. In gauge/gravity duality, quasinormal modes are
normalizable gravity solutions denoting the quasi-particle states in a hot and
dense medium. The real part of the corresponding complex frequency $\omega$,
$Re(\omega)$, is interpreted as the effective mass and the imaginary part,
$Im(\omega)$, is interpreted as the decay width. Finally, we write the vector
field $A_{\alpha}$ in the gauge invariant electric field
$E_{\alpha}=p_{\alpha}A_{t}+\omega A_{\alpha}$. Thus, in the zero momentum
limit, the energy density reads

$\rho(z)=T_{00}(z)=\frac{z^{2}e^{-\phi(z)}}{2g_{5}^{2}}(|E_{\alpha}|^{2}+\frac{af^{2}(z)}{|\omega|^{2}}|E_{\alpha}^{\prime}|^{2}).$
(20)

It is not difficult to find from the eq. (20) that quasinormal frequencies
must be known to obtain the differential configuration entropy (8). Therefore,
the quasinormal modes must be solved.

First, we solve the equations of motion coming from eq.(16) for the Gauss-
Bonnet gravity with the metric (11). By taking the radial gauge $A_{z}=0$ and
taking into account plane wave solutions $A_{\mu}(z,x_{3},t)=e^{-i\omega
t+iqx_{3}}A_{\mu}(z,\omega,q)$, the equations of motion are as follows

$\displaystyle
A_{t}^{\prime\prime}-(\frac{1}{z}+\phi^{\prime})A_{t}^{\prime}-\frac{q}{f}(qA_{t}+\omega
A_{x_{3}})$ $\displaystyle=$ $\displaystyle 0,\quad\quad$ (21) $\displaystyle
A_{\beta}^{\prime\prime}+(\frac{f^{\prime}}{f}-\frac{1}{z}-\phi^{\prime})A_{\beta}^{\prime}+\frac{1}{af^{2}}(\omega^{2}-aq^{2}f))A_{\beta}$
$\displaystyle=$ $\displaystyle 0,$ (22) $\displaystyle
A_{x_{3}}^{\prime\prime}+(\frac{f^{\prime}}{f}-\frac{1}{z}-\phi^{\prime})A_{x_{3}}^{\prime}+\frac{\omega}{af^{2}}(qA_{t}+\omega
A_{x_{3}})$ $\displaystyle=$ $\displaystyle 0,$ (23) $\displaystyle\omega
A_{t}^{\prime}+aqfA_{x_{3}}^{\prime}$ $\displaystyle=$ $\displaystyle 0.$ (24)

where $\beta=(x_{1},x_{2})$ and the prime $(^{\prime})$ denotes the derivative
with respect to radial coordinate $z$. The vector field $A_{\mu}$ can be
written in gauge invariant electric fields defined by $E_{\mu}$:
$E_{\|}=\omega A_{x_{3}}+qA_{t}$ and $E_{\bot}=\omega A_{\beta}$, where
$E_{\|}$ indicates the propagation along the momentum direction and $E_{\bot}$
means the propagation perpendicular to the momentum direction. So the
corresponding electric field component equations read

$\begin{split}0=&E_{\bot}^{\prime\prime}+(\frac{f^{\prime}}{f}-\frac{1}{z}-\phi^{\prime})E_{\bot}^{\prime}+\frac{\omega^{2}-aq^{2}f}{af^{2}}E_{\bot},\\\
0=&E_{\|}^{\prime\prime}\\!\\!+\\!\\!(\frac{aq^{2}f^{\prime}}{\omega^{2}\\!\\!-\\!\\!aq^{2}f}\\!+\\!\frac{f^{\prime}}{f}\\!\\!-\\!\frac{1}{z}\\!-\\!\phi^{\prime})E_{\|}^{\prime}\\!\\!+\\!\\!\frac{\omega^{2}\\!\\!-\\!\\!aq^{2}f}{af^{2}}E_{\|},\end{split}$
(25)

One can easily find that the equations for both the longitudinal component and
the transverse component are singular for $z=(0,z_{h})$. To numerically solve
the above electric field equations, we need two boundary conditions. To impose
the boundary conditions at the event horizon, it is convenient to rewrite each
of electric field equations into an Schrödinger-like form by defining the
tortoise coordinate, $\partial_{r_{*}}=-af(z)\partial_{z}$, and performing the
Bogoliubov transformation,
$E_{\|}=e^{\mathcal{B}_{\|}/2}\psi_{\|}\,,E_{\bot}=e^{\mathcal{B}_{\bot}/2}\psi_{\bot}$
with $\mathcal{B}_{\|}=\log\left(aq^{2}f(z)-\omega^{2}\right)+\phi(z)+\log(z)$
and $\mathcal{B}_{\bot}=\phi(z)+\log(z)$. The Schrödinger-like equation is
given by

$\begin{split}\partial^{2}_{r_{*}}\psi_{\|}+(\omega^{2}-U_{\|})\psi_{\|}&=0,\\\
\partial^{2}_{r_{*}}\psi_{\bot}+(\omega^{2}-U_{\bot})\psi_{\bot}&=0,\end{split}$
(26)

with the effective potential

$\begin{split}U_{\|}=&\frac{1}{4}(a^{2}f(z)^{2}\left(\frac{aq^{2}f^{\prime}(z)}{aq^{2}f(z)-\omega^{2}}+\phi^{\prime}(z)+\frac{1}{z}\right)^{2}+a^{2}f(z)\\\
&\left(4q^{2}-2f^{\prime}(z)\left(\frac{aq^{2}f^{\prime}(z)}{aq^{2}f(z)-\omega^{2}}+\phi^{\prime}(z)+\frac{1}{z}\right)\right)+2a^{2}\\\
&f(z)^{2}(\frac{aq^{2}\left(f^{\prime\prime}(z)\left(\omega^{2}-aq^{2}f(z)\right)+aq^{2}f^{\prime}(z)^{2}\right)}{\left(\omega^{2}-aq^{2}f(z)\right)^{2}}+\frac{1}{z^{2}}\\\
&-\phi^{\prime\prime}(z))-4(a-1)\omega^{2}),\\\
U_{\bot}=&\frac{1}{4z^{2}}(2a^{2}zf(z)\left(2q^{2}z-f^{\prime}(z)\left(z\phi^{\prime}(z)+1\right)\right)+a^{2}f(z)^{2}\\\
&\left(-2z^{2}\phi^{\prime\prime}(z)+z^{2}\phi^{\prime}(z)^{2}+2z\phi^{\prime}(z)+3\right)-4(a-1)\omega^{2}z^{2}),\end{split}$
(27)

At the event horizon $U_{\|/\bot}(z_{h})=0$, one can anlytically solve eq.(26)
and get $\psi_{\|/\bot}(z_{h})=c_{1}e^{-i\omega r_{*}}+c_{2}e^{+i\omega
r_{*}}$, which corresponds to the superposition of incoming and outgoing
waves. Because nothing can escape from the black hole, only the incoming wave
solution is physically allowed. Therefore, at $z\rightarrow z_{h}$, the
incoming wave solution for Schödinger-like equation (26) has the following
expansion:

$\psi_{\aleph}=e^{-i\omega
r_{*}}(1+a_{\aleph}(z-z_{h})+b_{\aleph}(z-z_{h})^{2}),$ (28)

with

$\begin{split}a_{\aleph}=&\frac{1}{M\omega^{4}z_{h}^{4}}(8a\omega^{2}\text{sech}^{2}\left(\frac{1}{Mz_{h}}-\frac{\kappa}{\sqrt{\Gamma}}\right)-4aMz_{h}\\\
&\left(8aq^{2}\delta_{\aleph\|}+\omega^{2}\left(4\kappa^{2}z_{h}^{2}+2Mz_{h}+q^{2}z_{h}^{2}+2\right)\right)),\end{split}$
(29)

where the formula of $b_{\aleph}$ is too long and will not be shown here and
$\delta_{\aleph\|}$ is the Kronecker delta function with $\aleph=(\|,\bot)$.

Then the incoming wave boundary condition can be obtained by the inverse
Bogoliubov transformation as follows:

$\begin{split}E_{\aleph}(z_{h})=&e^{-i\omega
r_{*}(z_{h})+\frac{\mathcal{B}_{\aleph}(z_{h})}{2}},\\\
E^{\prime}_{\aleph}(z_{h})=&(-i\omega
r^{\prime}_{*}(z_{h})+\frac{\mathcal{B}^{\prime}_{\aleph}(z_{h})}{2}+a_{\aleph})E_{\aleph}(z_{h}).\end{split}$
(30)

Next, one can use eq.(30) as boundary conditions to numerically solve eq.(25)
and search for complex frequencies $\omega$ making $E(0)=0$. The real part of
the corresponding quasinormal frequency is the effective mass, the imaginary
part gives the decay width, and the corresponding solution is the quasinormal
modes denoting the heavy quarkonium.

We can now turn to the differential configuration entropy. First of all, we
perform a Fourier transform on the energy density $\rho(z)$ in the coordinate
$z$ and split the energy density $\rho(k)$ as follows

$\displaystyle\rho(k)$ $\displaystyle=$
$\displaystyle(C(k)-iS(k))/\sqrt{2\pi},$ $\displaystyle C(k)$ $\displaystyle=$
$\displaystyle{\int_{0}}^{z_{h}}\rho(z)\cos(kz)dz,$ (31) $\displaystyle S(k)$
$\displaystyle=$ $\displaystyle{\int_{0}}^{z_{h}}\rho(z)\sin(kz)dz,$

with the modal fraction,

$\epsilon(k)=\frac{S^{2}(k)+C^{2}(k)}{[S^{2}(k)+C^{2}(k)]_{max}}.$ (32)

So the differential configuration entropy reads

$S_{DCE}=-\int_{-\infty}^{\infty}\epsilon(k)\log[\epsilon(k)]dk.$ (33)

##  IV  The Results of effective potential

Figure 2:  The effective potential as a function of the tortoise coordinates
with varying temperatures. From top to bottom, the curve sequentially
represents $T=0.2,0.3,0.4,0.5\,\text{GeV}$.

Figure 3:  The effective potential as a function of the tortoise coordinates
with varying energies at $\lambda_{GB}=0.05$ (top panel) and
$\lambda_{GB}=-0.05$ (bottom panel). The red, blue, magenta, green curve
sequentially denotes $\omega^{2}=20,40,60,80\,\text{GeV}^{2}$.

Let us start by studying the effective potential for vanishing wave number
$q=0$, for which the effective potential $U_{\|}$ and $U_{\bot}$ in Eq.(27)
have the same form,

$\begin{split}U=&U_{\|}|_{q=0}=U_{\bot}|_{q=0}=\frac{1}{4z^{2}}\\\
&(-2a^{2}zf(z)f^{\prime}(z)\left(z\phi^{\prime}(z)+1\right)+a^{2}f(z)^{2}\\\
&\left(-2z^{2}\phi^{\prime\prime}(z)+z^{2}\phi^{\prime}(z)^{2}+2z\phi^{\prime}(z)+3\right)\\\
&-4(a-1)\omega^{2}z^{2}).\end{split}$ (34)

One can find that the behavior of effective potential caused by higher-order
curvature corrections is determined by the last term $(1-a)\omega^{2}$, which
results in $U(z_{h})\neq 0$ at the horizon for $a\neq 1$ or $\lambda_{GB}\neq
0$ and the effective potential depends on the energy $\omega$. The effective
potential as a function of the tortoise coordinates with varying temperature
is shown in Fig.2. The potential shape behaves as a well which suggests some
quasiparticle states in the dual theory exist in the range of temperature,
corresponding to the bottomonium in the hot QGP. The depth of potential well
decreases with the increase in temperature, indicating the dissociation effect
of bottomonium increases when the temperature increases. In addition, near the
boundary($r_{*}=0$), there exists an infinite barrier localized which starts
from negative potential.

The appearance of higher-order curvature corrections also causes the effective
potential depending on the squared energy $\omega^{2}$. The effect of squared
energy on effective potential is related to the temperature parameter $a$.
When $a\leq 1$ ($\lambda_{GB}\geq 0$), the depth of potential well increases
as the squared energy increases, which means the probability of bottomonium
formation increases with the increasing energy, as displayed in the upper
panel of Fig.3. When $a\geq 1$ ($\lambda_{GB}\leq 0$), the depth of potential
well decreases as the squared energy increases, which indicates the
dissociation effect of bottomonium increases with the increasing energy, as
displayed in the lower panel of Fig.3.

In Fig.4, we plot the dependence of effective potential as a function of
tortoise coordinate on the constant $\lambda_{GB}$. An interesting phenomenon
is, as increasing constant $\lambda_{GB}$, the potential well becomes deeper
for zero wave number, corresponding to the probability of bottomonium
formation being larger, while that becomes shallower for finite wave number
indicating the melting effect being stronger. This behavior looks very
unreasonable. Because the large $\lambda_{GB}$ means the coupling strength
between plasma and heavy quarks is larger, which indicates the dissociation
effect is stronger for large $\lambda_{GB}$. We explain this non-trivial
behavior by the interplay of the interaction between the two heavy quarks and
the interaction between the medium with each of the heavy quarksZhao and Hou
(2022). One can well understand that a $Q\bar{Q}$ bound state denotes a
balance between repulsive kinetic( by an entropic effect) and attractive
potential energy(by an internal energy). Color screening modifies the
attractive binding energy between the quarks, while the increasing entropy
with large separation gives rise to a growing repulsionSatz (2015). For
smaller wave numbers, the interaction(binding) is stronger between quark and
anti-quark. However, for larger wave numbers, the interaction(dissociation) is
stronger between quark/anti-quark and mediums. The non-trivial behavior
suggests that here a two-body potential approach is no longer applicable, so a
kind of method related to an enhanced collective entropic quarkonium
dissociation may be usedKharzeev (2014), such as configuration entropy. Notes
that, at $\lambda_{GB}<0$, the effective potential at the horizon $U(z_{h})$
is less than $0\text{GeV}^{2}$, however, which is greater than
$0\text{GeV}^{2}$ for $\lambda_{GB}>0$ .

Figure 4:  The effective potential as a function of the tortoise coordinates
with different $\lambda_{GB}$ at zero wave number and finite wave number. The
red, blue, magenta, and green curve sequentially denotes
$\lambda_{GB}=-0.19,-0.09,0.0,0.09$ on the upper($q=0.0\,\text{GeV}$),
middle($q=4.0\,\text{GeV}$ for the parallel case), and
lower($q=4.0\,\text{GeV}$ for the perpendicular case) panels.

##  V  The Results of QNMs and differential configuration entropy

Figure 5:  The real part of the quasinormal frequencies as a function of the
temperature $T$ with different constants $\lambda_{GB}$. From bottom to top,
the curve represents $\lambda_{GB}=-0.19,-0.05,0.02,0.08$, respectively.
Figure 6:  The imaginary part of the quasinormal frequencies as a function of
the temperature $T$ with different constants $\lambda_{GB}$. From bottom to
top, the curve represents $\lambda_{GB}=-0.19,-0.05,0.02,0.08$, respectively.
Figure 7:  Quasinormal spectrum of electric field fluctuations in the plane of
complex $\omega$ defined by Eq.(25). From left to right, the curve represents
$\lambda_{GB}=-0.19,-0.05,0.02,0.08$, respectively. Figure 8:  The dependence
of the differential configuration entropy on temperature with varying
constants $\lambda_{GB}$. From bottom to top, the curve represents
$\lambda_{GB}=-0.19,-0.05,0.02,0.08$, respectively.

Quasinormal spectra (QNMs) is widely used in different research fields to
study different physics. In holography, research of QNMs became a standard
tool in treating the near-equilibrium problem of gauge theory. It gives the
location of the poles of the retarded correlators in the gauge theory, which
provides worthwhile information about the quasiparticle spectra function and
some transport coefficients.

Here, we consider the heavy mesons at rest in the hot medium. The real part of
the quasinormal frequencies is interpreted as the effective mass and the
imaginary part describes the decay width. In Fig.5, we plot the dependence of
effective mass on the temperature at four different values of the constant
$\lambda_{GB}$ and the decay width is presented in Fig.6. One can find that
the effective mass monotonically decreases with temperature in the lower
temperature region, while an opposite behavior is obtained in the higher
temperature region, which is consistent with Ref. Zhao and Hou (2022); Braga
and Da Mata (2020). In addition, the effective mass increases with the
increase of constant $\lambda_{GB}$. As expected, Fig.6 shows the decay width
(the reciprocal of lifetime) decreases with reduced temperature and the
dissociation effect becomes stronger with the increase of $\lambda_{GB}$. The
QNMs reveal the relation between effective mass and lifetime of quasiparticle
state in different temperature ranges, as displayed in Fig.7. For each curve,
the temperature increases from bottom to top. One can find that the dependence
of quasi-particle lifetime on effective mass is stronger and the strength of
the dependence is basically independent of constant $\lambda_{GB}$ in the
lower temperature region. However, in the higher temperature region, the
sensitivity of quasi-particle lifetime on effective mass increases with the
increase of constant $\lambda_{GB}$.

Finally, we calculate the DCE in function of temperature with different
constant $\lambda_{GB}$ in Fig.8. The DCE becomes bigger with the increasing
temperature, suggesting the dissociation effect being stronger, which is the
same as Ref.Braga and da Mata (2020a). Larger $\lambda_{GB}$ also increases
the DCE in the higher temperature region, this result is reasonable since
larger $\lambda_{GB}$ (or smaller $\eta/s$) in general means stronger
coupling, which implies that the interaction between quark/anti-quark and
medium is stronger than that between quark and anti-quark. In other words,
larger $\lambda_{GB}$ should have a stronger dissociation effect corresponding
to a larger DCE. However, the DCE basically remains unchanged with varying
temperature and $\lambda_{GB}$ in the lower temperature regions, which means
the existence of bottomonium is not affected by the thermal medium.

##  VI  Conclusion and discussion

In this letter, we study the effective potential, quasinormal spectra (QNMs)
and the differential configuration entropy (DCE). The results show that the
effective potential is not a good physical quantity to estimate the
dissociation strength of bottomonium under the very strong coupling QGP. At
the moment, a kind of method related to an enhanced collective entropic
quarkonium dissociation may be used, such as configuration entropy. By
calculating the QNMs, we find the behavior of effective mass as a function of
temperature is non-monotonic and the effective mass is larger for strong
coupling plasma. In particular, the results from QNMs and DCE indicate both
higher temperature and stronger coupling promote the dissociation effect,
leading to the suppression of the final yield from dilepton pair.

With some interesting conclusions obtained from this paper, we are also
curious about the properties of $J/\Psi$ under the strongly coupled plasma. We
will do this job in the future. In addition, we guess the effect of strong
coupling may affect the heavy vector mesons’ spectra, which is directly
related to the production rate of dilepton. One can calculate the current-
current correlation function of heavy quark to obtain it and compare the
result with the experiment, which may give a surprising conclusion. Of course,
there are also many other physical phenomena in experiments that urgently
require a self-consistent theory to explain, such as spin alignment. One can
also study the effect of rotating QGP on the QNMs and the DCE, as demonstrated
in Ref.Zhao et al. (2023).

##  VII acknowledgments

This work is supported in part by the National Key Research and Development
Program of China under Contract No. 2022YFA1604900. This work is also partly
supported by the National Natural Science Foundation of China (NSFC) under
Grants No. 12275104, No. 11890711, No. 11890710, and No. 11735007, as well as
by the Fundamental Research Funds for the Central Universities (Innovation
Funding Projects) 2020CXZZ108.

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