# A 1V 5-bits Low Power Level Crossing ADC with OFF state in idle time for
bio-medical applications in $0.18\mu m$ CMOS

Lucas Moura Santana, Duarte Lopes de Oliveira, and Lester de Abreu Faria
FootnoteL. M. Santana, D. L. de Oliveira and L. A. Faria are within the
Technological Institute of Aeronautics.Manuscript written on April 6, 2019;

###### Abstract

The ubiquitous use of sensing and signal processing is increasing
exponentially with the advance of the Internet of Everything (IoE). In this
context, the design of every time more power efficient sensor nodes is a must.
Within these nodes, one of the most power-hungry components are the analog-to-
digital converters (ADC). These components are used everywhere to translate
real-world analog signals into computer intelligible digital signals. One of
the promising architecture for the sensing of physiological signals is the
level crossing ADC due to the sparse characteristics of those signals. One of
the challenges to improve the power efficiency of this type of ADC lies in the
use of continuous comparators to keep track of the input signal within the
voltage references. The aim of this work is to investigate the impact of using
continuous comparator which can be turned off without incurring error to the
conversion of the level crossing ADC. New boundaries will be set for the
correct behavior of the level crossing ADC together with the conditions for
power saving with the proposed architecture. A 1V 5-bits level crossing ADC
was implemented using the TSMC $0.18\mu m$ process and fabricated for
laboratory measurements. The ADC consumes $12.2\mu W$ during tracking state
and with the proposed technique, the reduction of the average power can go
from 4.2% to 45.5% depending on the activity and the type of the input signal.

###### Index Terms:

Level crossing ADC, bio-medical, low power.

##  I Introduction

The data compression method using level crossing shows some interesting
features since it was proposed in the construction of medium resolution (4 to
10 bits) and low bandwidth Analog-to-Digital converters (ADC) [1, 2]. From the
performance perspective, the level crossing ADC has the advantage of using the
least amount of data to provide the output, being equal to two bits, one for
the direction of the level crossing and another for the occurrence of a level
crossing. Those two bits combined with an initial condition are able to
provide any number of bits at the output, given the proper behavior of the ADC
when tracking the input signal. Also the asynchronous behavior of the
sampling, given it is input dependent, indicates the level crossing ADC as a
good solution for the conversion of sparse signals, which are signals with
high spectrum spike contents followed by long idle periods [2].

Most of the recent work on level crossing ADCs are being made to lower the
power consumption of such converter while keeping the medium resolution to use
it as a feasible solution for bio-medical applications ([3, 4, 5]). In this
work, a technique to further reduce the power consumption under a known input
signal conditions is evaluated. This technique consists of turning off the
comparators for signal tracking while those are not being used. A set of
conditions for the application of such technique are given for the proper
functioning of the ADC under a well-behaved input signal. A prototype 5-bits
level crossing ADC is fabricated in $0.18\mu m$ to prove the operation of the
conversion. The final ADC consumes a tracking power of $12.2\mu W$ and such
power can be reduced by a 45.5% for a pure sine wave input and between 4.2% to
32.7% for a scaled ECG signal.

This work is then divided as follow for further explanation. Section II is
presenting the level crossing architecture implemented. Section III is
describing the design of the key components such as comparators and the finite
state machine, together with the implementation of the power reduction
technique. Section IV is about the measurements of the fabricated ADC and the
analysis of the power reduction technique. Finally, Section V concludes the
paper and presents further steps to improve the ADC power reduction.

##  II Architecture Review

To understand the behavior of the general level crossing ADC one can
approximate this type of conversion to the one of two other types of ADC: the
flash and the Successive Approximation Register (SAR). The level crossing is
similar to the flash since it continuously generates voltage references that
feed comparators to decide the actual conversion output of the ADC. The
similarity with the SAR is in the aspect of the sampling event triggering a
binary search, that is, every time a conversion is being made the system will
take action to update its values if the level crossing is going up or if it is
going down.

The implementation of level crossing ADCs is done mainly by two different
architectures: the floating window and the fixed window [6]. A window may be
seen as the space between two reference voltages where the input signal is
located at a given time. The event of the signal going out of the window
triggers a level crossing, thus the update of the system to accommodate the
window to the input signal again. The difference in voltage between the
references is always set to be one quantization step of the ADC or one LSB.
For its simplicity and robustness, the floating window is used in this work
and it is further explained. Figure 1 shows the simplest way to implement the
floating window level crossing architecture and it depicts the moving window
tracking the input signal.

(a) Floating window generic implementation

(b) Floating window tracking with the moving voltage references

Figure 1: Floating window architecture and tracking of the input signal.

As for choosing the implementation of the components presented in the
architecture, there lie the many variations of the level crossing ADCs. For
the DACs, the work in [3] uses a resistive ladder for its flexibility in
generating the moving window; [4] uses an innovative 1-bit CDAC which is able
to perform with great efficiency in the fixed window scheme; [5] implements a
set of analog memories to add the capability to easily change the resolution
of the ADC accordingly. For the logic that keeps track of the crossing event,
there are also many ways to control RTL components such as registers and ALUs.
However, when reaching the comparator design phase, one could only choose
among the topologies within the continuous comparators. When compared with the
widely used dynamic comparators, the continuous type is a much more energy
demanding component, although it provides the capability to be always tracking
the input changes [7].

The work in [5] tries to use dynamic comparator as means to keep track of the
input signal, but it turns the level crossing architecture into a clocked
system which gives away on the purpose of being efficient in the conversion of
sparse signals. Nonetheless, the improvement in power efficiency in that work
using the nominal voltage of the technology is noteworthy.

This work goes further in the investigation of adapting the comparators as a
means to reduce the average power consumption. For that matter, it was used
continuous comparators that could be turned off at given time so the static
power is minimized. To enable this control over the comparator, the modified
floating window architecture in Figure 2 was implemented.

One can understand a level crossing event in this architecture as follow. At
first, the signal $V_{IN}$ is within the window formed by $V_{+}$ and $V_{-}$,
then suppose an upper level crossing occurred, that is, the signal $V_{IN}$
became bigger than $V_{+}$. This will trigger the signal INC which feeds the
Burst Mode Asynchronous Finite State Machine (BM AFSM). This state machine
immediately turns off both comparators and proceeds to update the datapath and
register holding the ADC conversion level at the moment. The changes in the
register will update the value of the DACs, in this case for an upper output
value, and the floating window moves up as it was supposed to be. The timing
for the update of the ADC is set by the communication signals REQ and ACK on
the interface with the state machine. During this whole update phase, the
comparators will be off, so maximizing the time they are off will provide
energy savings. However, the comparators cannot be definitely off since they
need to be active before the next expected level crossing, thus the boundaries
for the off time need to be set and the energy saving will be limited by it.
Figure 3 show the time representation of the comparator states and the events
within the level crossing ADC. Note the time to update the digital components
is much smaller than the time of decision from the comparator (timing in the
figure is not scaled).

Figure 2: Proposed floating window architecture with added signal to turn off
the comparators. Figure 3: Timing of the events within two expected level
crossing in the ADC.

##  III ADC design

###  III-A Comparator

The comparators implemented in this work are based on the works of [3] and
[8]. The self-bias comparator proved its low power performance in [3]. This
component is composed of three stages, the first is a rail to rail
differential input pair with folded cascode load. The second stage is a
differential pair with a cross-coupled load to increase the overall gain of
the comparator. Finally, the third stage is a modified positive feedback latch
used to add the last bits of gain and to convert the differential signal to a
single ended. Analytic gain equations could be taken for the stages, but with
a moving bias point, it is hard to accurately calculate the gain of the
comparator without a simulation tool. However, general amplifiers guidelines
([7]) are used to size the transistors and the final result is within the
expected performance.

The main difference is with respect to the measures taken to provide the
turning off capability. Figure 4 provides the schematic for the three stages
of the comparator. Note the main difference with the original comparator are
the transistors $M_{N8}$ and $M_{N15}$, which both serve as discharging
transistors to effectively prepare the comparator to be turned on again
without causing glitches at the output. The signal $ON_{comp}$ is a buffered
version of the ON signal from the BM AFSM, in a way it can provide enough
current to bias the comparator, and it also has the function of a voltage
supply for the stages of the comparator.

(a) First stage

(b) Second stage

(c) Third stage

Figure 4: Three stages of the self biased comparator used to track the input
signal.

Post-layout simulations show the typical decision time for the comparator to
be around 659.5ns. With such value in hands, it is possible to calculate the
expected bandwidth of the level crossing ADC using the Bernstein Theorem and
the Equation 1 derived from [9]. Where $\Delta$ is equal to one quantization
level and A is equal to half of the full scale voltage. This gives rise to a
bandwidth of 11kHz for an ADC working at the full scale of 1V with such
comparator.

$f_{max}=\frac{\Delta}{A\times 2\pi\times t_{comp}}$ (1)

Note this is not the equivalent of a Nyquist frequency since this is not a
uniform sampled environment. In fact, the bandwidth here will be the maximum
frequency up to which the ADC can still reliably track the input signal at
full scale.

###  III-B Digital components

The main concern in the digital part of the ADC is the state machine. This
machine was chosen to be designed using the burst mode asynchronous paradigm
[10]. This paradigm is meant for monotonic signal transitions in event-based
environments, such as the one presented in the level crossing scheme of this
work. The graph of the state transitions describing the behavior of the
signals is shown in Figure 5. The “$+$” sign refers to a signal transitions
from logic 0 to logic 1 and a “$-$” sign refers to a signal transitions from
logic 1 to logic 0. The signals INC, DEC, and ACK are inputs for the state
machine, while the signals ON, REQ, SEL and L are the output. State 0 is the
tracking state where the comparators are active. The path $0\rightarrow
1\rightarrow 3\rightarrow 0$ is triggered by a falling level crossing and the
path $0\rightarrow 2\rightarrow 3\rightarrow 0$ is triggered by a rising level
crossing. Note the time the ADC spends on the states 1, 2 and 3 contribute to
the power savings since the comparator are off while in those states.

Figure 5: Graph of the state transitions of the BM AFSM.

The state machine was synthesized using the 3D tool [11] in the feedback
output architecture, where the outputs of the state machine are used as state
variables, thus saving area and possibly power. The resulting Boolean
equations are shown in the set of Equations 2.

$\displaystyle REQ$ $\displaystyle=INC+DEC+\overline{ACK}.REQ$ (2)
$\displaystyle ON$ $\displaystyle=\overline{INC+DEC+ACK+REQ}$ $\displaystyle
SEL$ $\displaystyle=DEC+\overline{ACK}.SEL$ $\displaystyle L$
$\displaystyle=\overline{INC}.\overline{DEC}.ACK$

To interface with the behavior of the BM AFSM, the designed register is a
regular DFF based register working in the positive edge of the load signal L.
The datapath is a custom ripple carry adder where the SEL signal swaps the
second operand to provide an increment by 1 or to provide a decrease by 1
depending on what is needed. Note the standard value of SEL during track mode
gives the datapath an output of $+1$ over the conversion level output
(register value) and that is used to feed the DAC generating the $V_{+}$.

###  III-C DAC

The DACs generating the voltage references were chosen to be CDACs based on
charge sharing. This type of DAC has good performance and matching in medium
resolutions [12] and has virtually no static power consumption. The focus of
the design is given to the layout of such component. Based on the work of [13]
the floor planning was designed as shown in Figure 6 aiming to minimize the
routing parasitic and to increase the capacitors’ matching.

Figure 6: Layout of the capacitor banks of the CDAC.

###  III-D ACK generator

One of the simplest ways to provide the ACK signal to the BM AFSM in a
controlled manner is shown in Figure 7. The clock-gating scheme is meant to
decrease the power consumption during the self-transitions since the global
clock signal does not arrive at the flip-flop in that condition [14]. In
Figure 8, one can see the timing of the signal within the ACK generator and
further extract the energy savings that it adds to the system. First, we
consider a REQ+ signal just occurred, that means the comparators are off and
the BM AFSM is waiting for an ACK+ signal. This waited response happens in the
next rising edge of the input clock of the generator, taking the BM AFSM to a
state where it keeps waiting for the ACK– signal to go back to tracking mode.
This signal is given at the second rising edge after the REQ+ occurred. Note
that, even though a clock signal is being used for the ACK generator as means
to control the time the ADC will be with the comparators off, the converter
itself did not turn into a synchronous one, but rather it is using a clocked
communication.

Figure 7: Controlled ACK generator for the communication with the BM AFSM.
Figure 8: Timing for the behavior of the ACK generator after a REQ signal.

The energy saving estimate can be extracted from the fact that, given a big
number of level crossing, the distribution of the moment when REQ+ occurs
within a clock period becomes uniformly distributed. So the expected value
from the moment a REQ+ occurs to the moment an ACK– is received back is equal
to the expected value of the uniform distribution plus one clock period. This
time is the estimate of the time the comparators are off. With this value in
hand, the set of Equations 3 models the average power obtained by measuring
the power of the ADC for a long time ($t$).

$\displaystyle\hat{t}_{OFF}=\frac{3\times T_{clk}}{2}$ (3) $\displaystyle
T_{OFF}=N_{crossings}\times\hat{t}_{OFF}$ $\displaystyle
P_{ADC}^{mean}=P_{ADC}^{ON}\times(1-\frac{T_{OFF}}{t})+P_{ADC}^{OFF}\times(\frac{T_{OFF}}{t})$

The increase in the period of the clock leads to more time in an off state and
smaller average power. However, the value of this clock period is bounded by
the frequency content of the input signal following the Equation 4. This bound
comes from the fact that the ACK– signal has to arrive before the next
expected level crossing, thus the worst case level crossing has to be within
two clock periods to accommodate the communication time.

$T_{clk}\leq\frac{\Delta}{4\pi\times f_{IN}\times A}$ (4)

Modeling this behavior in a numerical calculation software such as Octave, it
is possible to plot the practical limits for power reduction with a set of
sine waves as the input. Figure 9 shows that for frequencies up to 1kHz, the
amount of time the ADC will be off during one period using the maximum allowed
clock period is close to 46.2%.

Figure 9: Plot of theoretical off time for different clock frequencies
(curves) and different input frequencies (x-axis).

For testing purposes, this component to generate the ACK signal for the ADC
was not embedded within the integrated circuit, but rather it was modeled
outside of the chip.

##  IV Measurements and Analysis

The layout for the integrated circuit is shown in Figure 10. It was fabricated
using the TSMC $0.18\mu m$ technology and it occupies an area of $0.05mm^{2}$.
For measurement purposes, the ADC was subject to a slow triangular wave with
limited amplitude and DC offset close to the middle voltage of conversion
($V_{DD}/2$). This was meant to ease the communication with the ADC and enable
the precise collection of the on and off state power. Figure 11 shows the
output waveform of the ADC under these conditions while Figure 12 shows a
magnified version of the power waveform of the measured ADC.

Figure 10: Layout of the ADC occupying an area of $0.05mm^{2}$. Figure 11:
Plot of the output of the measured ADC during one clock period. Figure 12:
Magnified plot of the measured ADC power consumption.

With the extracted performance of the ADC, it was possible to evaluate the
performance of the power reduction technique under the conversion of
physiological signal using the ECG database in [15] and the toolbox in [16].

Under an ECG signal boundary, the previous timing analysis was made to design
the clock period used to drive the system that generates the ACK signal to the
ADC. Then, the ADC model was subject to a full scale ECG and the off time was
measured with the modelling tool. A frame of such evaluation is shown in
Figure 13 with the darker signal showing the moments the ADC was turned off.
The average power reduction of the ADC was close to 4.2% for the whole frame,
and the peak reduction was close to 32.2% when the converter is subject to the
most activity, that is, when it is converting the QRS spike of the ECG signal.

Figure 13: Magnified plot of the estimated off time of the ADC overlapped with
the conversion of the ECG signal.

Although the level crossing ADC is meant for a low power behavior during idle
time in sparse signals, these results shows how the power consumption can be
greatly reduced during high activity moments of conversions, increasing the
usability of the level crossing ADC.

Table I shows the performance of different level crossing ADCs from the last
years. The bigger average power consumption of the fabricated ADC is due to
its increased bandwidth, which depending on the application could be reduced
to further save energy. Also, the reduction in power in [3] and [4] is boosted
by the reduction of the nominal voltage, what does come with the penalty of
reduced dynamic range.

TABLE I: Performance comparison of recent level crossing ADCs Reference | Bits | Window | Technology | $V_{DD}$ | Power | Bandwidth
---|---|---|---|---|---|---
[17] | 10 | Floating | $0.15\mu m$ | $1.8V$ | $1300\mu W$ | $5kHz$
[4] | 6 | Fixed | $0.18\mu m$ | $0.8V$ | $0.5\mu W$ | $5kHz$
[3] | 5 | Floating | $0.13\mu m$ | $0.3V$ | $0.2\mu W$ | $1kHz$
[5] | 4-8 | Floating* | $0.35\mu m$ | $1.8V$ | $2\mu W$ | $1kHz$
This work | 5 | Floating | $0.18\mu m$ | $1.0V$ | $6.7\mu W-12.2\mu W$ | $1kHz$
*synchronous level crossing

##  V Conclusion

The proposed technique to reduce the power of an asynchronous level crossing
ADC by means of turning off the comparators during idle time was implemented
and showed potential in reducing the average power by maximizing the off time
of the ADC. Simulations showed a limit of 45.5% of power reduction for the
conversion of pure sine waves and values smaller than that for ECG scaled
waves, being the actual reduction a function of the activity of the signal. It
was noted that the system still has margin to power savings by redesigning the
comparator for smaller bandwidths and reducing the supply voltage in
environments that allows reduced dynamic range. As for the next steps on the
research, new topologies of comparators should be investigated to enable an
easier control of the on/off time of the ADC with smaller static power
consumption, such that the proposed technique could be ported to either
floating window and fixed window level crossing ADCs with the lesser effort.

## Acknowledgment

The authors would like to thank professor Jacobus Swart and IMEC for making
the fabrication of the IC possible through the program Mini-ASIC. This study
was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) - Finance Code 001”

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Lucas Moura Santana was born in Redenção, Brazil. He received his B.Sc degree
in electronic engineering from the Technological Institute of Aeronautics -
ITA, in 2017. Currently he is pursuing his M.Sc. degree in the area of low
power analog-to-digital converters focused on bio-medical applications. His
research interests are ADC, low power circuits, processing of physiological
signals and synthesis of asynchronous circuits.
---
Duarte de Oliveira Lopes was born in Vila Real, Portugal. He received the
B.Sc. degree in electrical engineering from the University of Mogi das Cruzes,
Brazil, in 1980, the M.Sc. degree in computer science from the Technological
Institute of Aeronautics, SJC, São Paulo, Brazil, in 1988, and the Ph.D.
degree in electrical engineering from the Polytechnic School of the University
of São Paulo in 2004. Since 2013, he is an Associate Professor in the
Electronic Engineering Division at the Technological Institute of Aeronautics
- ITA, Brazil. His research interests include asynchronous circuits, computer-
aided digital design, low power digital systems, and logic synthesis. He is
the author of more than 100 technical papers.
---
Lester de Abreu Faria was born in Rio de Janeiro - RJ, Brazil. He has received
degrees in aeronautical sciences from the Brazilian Air Force Academy, in
1993, and in electronic engineering from the Technological Institute of
Aeronautics-ITA, in 2004. He received the M.S. degree in electronic
engineering and computer science, in 2010, with emphasis on microelectronics,
testability and design of analog read-out integrated circuits (ROICs). He
received the Dr.Sci. degree in cryogenic integrated circuits for infrared
detectors applications in 2014. He is currently a Pilot Colonel in the
Brazilian Air Force and an Associate Professor of the Electronic Engineering
Division, in Technological Institute of Aeronautics. His research interests
include read-out integrated circuits for infrared detectors, cryogenic models
for conventional MOS transistors, rad-hard components and synthesis of
asynchronous circuits.
---


# Rotation effect on the spectral function of heavy vector mesons
in holographic QCD

Xiao-Long Wang  College of Science, China Three Gorges University, Yichang
443002, China  Sheng-Qin Feng  Corresponding author<EMAIL_ADDRESS>College of Science, China Three Gorges University, Yichang 443002, China
Center for Astronomy and Space Sciences and Institute of Modern Physics, China
Three Gorges University, Yichang 443002, China  Key Laboratory of Quark and
Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China

###### Abstract

Abstract: Exploring heavy vector mesons of the $J/\psi$ and $\Upsilon(1S)$ is
crucial for understanding the quark gluon plasma (QGP) formed in heavy ion
collisions. The influences of rotational effect on the properties of the
$J/\psi$ and the $\Upsilon(1S)$ are investigated by incorporating rotation
medium into the holographic QCD. It is found that temperature, chemical
potential, and rotational radius effects enhance the dissociation process of
the $J/\psi$ and the $\Upsilon(1S)$ states within the medium. This rotation-
induced effect is more significant for heavy vector mesons in the transverse
direction than that of the longitudinal direction. The first holographic study
on the influence of the radius of a homogeneous rotating system on the vector
meson spectrum is proposed. It is found that increasing in rotation radius
promotes the dissociation of vector mesons of the $J/\psi$ and $\Upsilon(1S)$.
We also find that the dissociation perpendicular to the direction of
rotational angular velocity is more significant than that parallel to it at
large rational radius.

##  I Introduction

As is well known, relativistic heavy ion collisions produce a high-
temperature, high-density medium, known as quark-gluon plasma (QGP) Adcox  _et
al._ (2005); Adams  _et al._ (2005); Arsene  _et al._ (2005). In this QGP
state, gluons and light quarks are in a deconfinement phase, exhibiting
perfect fluid behavior that is approximately thermalized, providing an
important tool for exploring quantum chromodynamics (QCD). These QGP matters
are mutually correlated through strong interactions, but they are not confined
within hadrons. They have an extremely short-lived $(\sim 5-10fm/c$), and
direct observation of the plasma is impossible. Heavy Mesons composed of $c$
(charm) or $b$ (bottom) quarks are more likely to survive long time in the hot
QGP medium compared to mesons composed of lighter quarks : $u$ (up), $d$
(down) and $s$ (strange). It is believed that heavy mesons can serve as probes
to indirectly gather information about the Quark-Gluon Plasma (QGP) Matsui and
Satz (1986); Ma (2019); Karsch (2005).

Due to the influence of temperature, chemical potential, and other relevant
properties on the properties of QCD media, these heavy vector mesons undergo
dissociation in high-temperature media. Therefore, understanding the thermal
behavior of heavy vector mesons, especially the relationship between their
dissociation degree and medium dynamics, is crucial for us to have a deeper
understanding of the properties of QCD media.

In relativistic heavy ion collisions, in addition to the presence of strong
magnetic fields Guo  _et al._ (2019); She  _et al._ (2018); Zhong  _et al._
(2014); Bzdak and Skokov (2012); Deng and Huang (2012), the plasma also
exhibits significant rotational vortex characteristics. Some studies have
introduced rotation into holographic research Wang and Feng (2024); Braga  _et
al._ (2022); Bravo Gaete  _et al._ (2017); Erices and Martinez (2018); Braga
and Ferreira (2023); Zhao  _et al._ (2023) and effective field theory Sun and
Huang (2022); Jiang and Liao (2016a); Qiu and Feng (2023); Qiu  _et al._
(2023); Bao and Feng (2024); Zhu and Feng (2023). The phases of matter and the
properties of associated particles become very complex under rotation, which
has recently attracted great interest Wang and Feng (2024); Sun and Huang
(2022). These studies are closely related to the properties of strong
interactions in quantum chromodynamics (QCD). For example, astrophysical
objects such as neutron stars composed of dense QCD matter can rotate rapidly
Watts  _et al._ (2016); Grenier and Harding (2015). The QCD medium produced by
non-central nucleus-nucleus collisions also carries a large amount of
rotational angular momentum, with an order of magnitude of
$10^{4}-10^{5}\hbar$, and local rotational angular velocity can reach the
order of magnitude of 0.01-0.1 GeV Liang and Wang (2005); Huang  _et al._
(2011); Becattini  _et al._ (2008); Jiang and Liao (2016b); Pang  _et al._
(2016); Csernai  _et al._ (2013); Deng and Huang (2016). On the other hand,
significant progress has also been made in simulating rotating QCD matter
using lattice gauge theory Yamamoto and Hirono (2013).

The gauge/gravity duality has become an important research tool for exploring
QCD phase transitions Wang and Feng (2024); Deng and Feng (2022); Chen  _et
al._ (2018, 2021) and transport properties Finazzo  _et al._ (2015); Li  _et
al._ (2015); Giataganas  _et al._ (2018). Recently, holographic methods have
been widely used to study the properties of heavy flavor spectral functions
under strong magnetic fields and medium rotation conditions Karch  _et al._
(2006); Vega and Martin Contreras (2019); Martin Contreras  _et al._ (2021a);
Mamani  _et al._ (2022); Fujita  _et al._ (2010); Satz (2006); Braga and
Ferreira (2019a, b); Martin Contreras  _et al._ (2021b); Zöllner and Kämpfer
(2021); Chernodub (2021); Zhu  _et al._ (2019); Feng  _et al._ (2020). The
gauge/gravity duality has not only become a valuable tool for exploring QCD
phase transitions and transport properties, but also been widely used to
explore the properties of heavy meson spectral functions.

In this article, we will use the holographic model to study the rotating QCD
matter, where the rotating AdS black hole is the holographic dual of the
plasma. The heavy vector meson spectral function of the rotating QCD system by
introducing a metric with rotating cylindrical coordinates is studied.
Although there are many non-perturbative methods to explore the heavy vector
meson spectral function Zhao and Hou (2023); Zhou  _et al._ (2021), the main
purpose of this article is to study the influence of plasma on the heavy
vector meson dissociation under rotation and finite chemical potential. We
focus on studying the dependence of the dissociation process of heavy vector
mesons $J/\psi$ and $\Upsilon(1S)$ on temperature, chemical potential, and
rotation radius.

Refenence Wang and Feng (2024) first used holography to study the influence of
the radius of a uniform rotating system on the QCD phase diagram. It is found
that when discussing the rotating system of QCD medium, the phase transition
characteristics strongly depend on the finite size of the rotating system. Due
to the cylindrical symmetry of the rotating system, the rotation radius
$\mathit{l}$ has become an important characteristic of the rotating system. We
will use holography to mainly discuss the influence of the radius of a uniform
rotating system on the heavy vector meson spectrum, and understand how the
finite size of the rotating system affects the characteristics of the QCD
system spectrum function in this article.

The structure of the paper is as follows: in Sec. II , we develop the
holographic model in plasma and extend it to finite temperature and finite
chemical potential. A holographic model of the spectral function of heavy
vector mesons in a rotating plasma is introduced In Sec. III. In Sec. IV , we
present and discuss the calculated results of spectral functions of the heavy
vector mesons. Finally, in Section V, we provide summary and conclusions.

##  II Holographic QCD theory under rotational background

The generalized version of the soft wall model Karch  _et al._ (2006) is
introduced, where the vector meson is described by a vector field
$V_{m}=(V_{\mu},V_{z})(\mu=0,1,2,3)$, which is dual to the current
$J^{\mu}=\psi\gamma^{\mu}\psi$ of the gauge theory. The standard Maxwell
action is

$S=-\int d^{4}xdz\frac{Q}{4}F_{mn}F^{mn},$ (1)

where $Q=\frac{\sqrt{-g}}{h(\phi)g_{5}^{2}}$,
$h(\phi)=e^{\phi(z)},F_{mn}=\partial_{m}V_{n}-\partial_{n}V_{m}$, and
$\phi(z)$ the dilaton background.

The metric is given by in the background of a charged black holes:

$ds^{2}=\frac{R^{2}}{z^{2}}(-f(z)dt^{2}+\frac{dt^{2}}{f(z)}+d\vec{x}\cdot
d\vec{x}),$ (2)

with

$f(z)=1-\frac{z^{4}}{z_{h}^{4}}-q^{2}z_{h}^{2}z^{4}+q^{2}z^{6},$ (3)

and

$\phi(z)=k^{2}z^{2}+Mz+\tanh(\frac{1}{Mz}-\frac{k}{\sqrt{\Gamma}}),$ (4)

where $k$ represents the quark mass, $M$ denotes a large mass and $\Gamma$ is
the string tension of the quark pair associated with the non-hadronic decays
of heavy quarkonium. There is a matrix element
$<0|J_{\mu}(0)|X(1S)>=\in_{\mu}f_{n}m_{n}$ (where represents the heavy mesons,
$<0|$ is the hadronic vacuum and $J_{\mu}(0)$ is the hadronic current and
$f_{n}$ is the decay constant). By fitting the mass spectrum, one determines
the best values Mamani  _et al._ (2022) for the three energy parameters in the
scalar field for charmonium and bottomonium as

$k_{c}=1.2GeV,\sqrt{\Gamma_{c}}=0.55GeV,M_{c}=2.2GeV,$ (5)
$k_{b}=2.45GeV,\sqrt{\Gamma_{b}}=1.55GeV,M_{b}=6.2GeV.$ (6)

The horizon position $z_{h}$ of the black hole is obtained by the condition
$f(z_{h})=0$. The temperature of the black hole is determined by the
requirement of the Euclidean version of the metric having no conical
singularity at the horizon. It satisfies the following relation:

$T=\frac{|f^{\prime}(z)|_{(z=z_{h})}}{4\pi}=\frac{1}{\pi
z_{h}}-\frac{q^{2}z_{h}^{5}}{2\pi},$ (7)

where the parameter $q$ is proportional to the black hole charge, which is
related to the quark chemical potential $\mu$. In the dual supergravity
description, the field $V_{0}$ acts as a source for the correlator’s
operators. Therefore, a specific solution of the vector field $V_{M}$ was
considered, where the non-zero component is given by:
$V_{0}=A_{0}(z)(V_{z}=0,V_{i}=0)$. Assuming that the relationship between $q$
and $\mu$ is the same as in the background-free case, i.e., $\phi(z)=0,$ the
solution for the time component of the vector field is given
by:$A_{0}(z)=c-qz^{2}$, where $c$ is a constant. By requiring $A_{0}(0)=\mu$
and $A_{0}(z_{h})=0$, one obtain:

$\mu=qz_{h}^{2}.$ (8)

In the early stages of non-central heavy ion collisions, the produced quarks
carry a large initial orbital angular momentum $J\propto b\sqrt{S_{NN}},$
where $b$ is the impact parameter and $\sqrt{S_{NN}}$ is the center-of-mass
energy of the nucleon-nucleon collision. Although most of the angular momentum
is carried away by the so-called ”spectators” during the initial collision
stage, a significant portion of the angular momentum remains in the resulting
quark-gluon plasma.

Basing on the study in Bravo Gaete  _et al._ (2017); Erices and Martinez
(2018); Zhou  _et al._ (2021); Braga and Ferreira (2018); Nadi  _et al._
(2019) , we extend the holographic QCD model to include the case of a rotating
black hole with a planar horizon. The general form of the metric of the static
coordinate system is given by:

$ds^{2}=-g_{tt}dt^{2}+g_{zz}dz^{2}+g_{x_{1}x_{1}}dx_{1}^{2}+g_{x_{2}x_{2}}dx_{2}^{2}+g_{x_{3}x_{3}}dx_{3}^{2}.$
(9)

Now, in order to analyze the case of a rotating plasma with uniform angular
velocity, we introduce an AdS5 space with cylindrical symmetry and describe
its structure by rewriting the metric as

$ds^{2}=-g_{tt}dt^{2}+g_{zz}dz^{2}+g_{\theta\theta}\mathit{l}^{2}d\theta^{2}+g_{x_{2}x_{2}}dx_{2}^{2}+g_{x_{3}x_{3}}dx_{3}^{2}.$
(10)

Then the angular momentum will be turned on in the angular coordinate $\theta$
through the standard Lorentz transformation. As in Refs. Martin Contreras  _et
al._ (2021a); Braga and Ferreira (2019b), we introduce rotation through a
Lorentz-like coordinate transformation as

$t\rightarrow\gamma(t+\Omega l^{2}\theta),$ (11)

and

$\theta\rightarrow\gamma(\Omega t+\theta),$ (12)

where $\gamma=\frac{1}{\sqrt{1-\Omega^{2}l^{2}}}$ is the usual Lorentz factor,
$\Omega$ is the angular velocity of rotation, and $l$ is the radius of the
rotating axis. With the introduction of rotation, the original metric of Eq.
(9) becomes the following form

$\displaystyle ds^{2}=$
$\displaystyle(g_{\theta\theta}\Omega^{2}l^{2}-g_{tt})\gamma^{2}dt^{2}+2\gamma^{2}\Omega
l^{2}(g_{\theta\theta}-g_{tt})dtd\theta$ (13) $\displaystyle\
+\gamma^{2}(g_{\theta\theta}-\Omega^{2}l^{2}g_{tt})l^{2}d\theta^{2}+g_{zz}dz^{2}+g_{x_{2}x_{2}}dx_{2}^{2}+g_{x_{3}x_{3}}dx_{3}^{2}.$

The Hawking temperature and chemical potential of a rotating black hole are
given by

$T=\big{(}\frac{1}{\pi
z_{h}}-\frac{q^{2}z_{h}^{5}}{2\pi}\big{)}\sqrt{1-\Omega^{2}l^{2}},$ (14)

and

$\mu=qz_{h}^{2}\sqrt{1-\Omega^{2}l^{2}}.$ (15)

##  III The spectral functions of heavy vector mesons

By making a variable substitution $l\theta=x_{1}$ with Eq. (10), one can
obtain a new metric representation along the $x_{1}$ direction

$ds^{2}=-G_{tt}dt^{2}+G_{x_{1}x_{1}}dx_{1}^{2}+G_{tx_{1}}dtdx_{1}+G_{x_{1}t}dx_{1}dt+G_{x_{2}x_{2}}dt_{2}^{2}+G_{x_{3}x_{3}}dt_{3}^{2}+G_{zz}dz^{2}.$
(16)

The spectral function of the charmonium and bottomonium can be calculated
using the membrane paradigm Zöllner and Kämpfer (2021). The equations of
motion obtained from Eq. (1) are as follows:

$\partial_{m}(QF^{mn})=\partial_{z}(QF^{zn})+\partial_{\mu}(QF^{\mu m}),$ (17)

where $F^{mn}=g^{m\alpha}g^{n\beta}F_{\alpha\beta},n=(0,1,2,3,4)$ and
$\mu=(0,1,2,3)$. For the $z$-direction, the conjugate momentum of the gauge
field $A^{\mu}$ is given by the following equation:

$j^{\mu}=-QF^{z\mu}.$ (18)

Now we consider the plane wave solution of vector field $A^{\mu}$ propagates
in the direction of $x_{1}$, and divide the equation of motion into a
longitudinal channel involving fluctuations along $(t,x_{1})$ and a transverse
channel involving fluctuations along the spatial directions $(x_{2},x_{3}).$
Combining Eq. (18) with Eq. (17), one can obtain the dynamical equation for
the longitudinal case as:

$-\partial_{z}j^{t}-\frac{\sqrt{-g}}{h(\phi)}\big{(}g^{x_{1}x_{1}}g^{tt}+g^{x_{1}t}g^{tx_{1}}\big{)}\partial_{x_{1}}F_{x_{1}t}=0,$
(19)
$-\partial_{z}j^{x_{1}}+\frac{\sqrt{-g}}{h(\phi)}\big{(}g^{tt}g^{x_{1}x_{1}}+g^{tx_{1}}g^{x_{1}t}\big{)}\partial_{t}F_{x_{1}t}=0,$
(20) $\partial_{x_{1}}j^{x_{1}}+\partial_{t}j^{t}=0.$ (21)

Using Bianchi’s identity, one can obtain:

$\partial_{z}F_{x_{1}t}-\frac{g_{zz}h(\phi)}{\sqrt{-g}}\partial_{t}\left[g_{x_{1}x_{1}}j^{x_{1}}+g_{x_{1}t}j^{t}\right]-\frac{g_{zz}h(\phi)}{\sqrt{-g}}\partial_{x_{1}}\left[g_{tt}j^{t}-g_{tx_{1}}j^{x_{1}}\right]=0.$
(22)

Now, it is possible to define the conductivity of the longitudinal channel
with respect to $z$

$\sigma_{L}(\omega,z)=\frac{j^{x_{1}}(\omega,z)}{F_{x_{1}t}(\omega,z)},$ (23)

and its derivative is defined as

$\partial_{z}\sigma_{L}(\omega,z)=\frac{\partial_{z}j^{x_{1}}}{F_{x_{1}t}}-\frac{j^{x_{1}}}{F_{x_{1}t}^{2}}\partial_{z}F_{x_{1}t}.$
(24)

The Kubo formula $\sigma(\omega)=iG_{R}(\omega)/\omega$ displays the five-
dimensional conductivity at the boundary, which allows us to define

$\sigma_{L}(w)=\frac{-G_{R}^{L}(w)}{iw}.$ (25)

Notice that the real part is $\textrm{Re}\sigma(\omega)=\rho(\omega)/\omega,$
where $\rho(\omega)\equiv-\textrm{Im}G_{R}(\omega)$ is the spectral function.
To obtain the flow equation Eq. (24), one assumes that
$A_{\mu}=A_{n}(p,z)e^{-i\omega t+ipx_{1}},$ where $A_{n}(p,z)$ represents the
quasi-normal mode. Therefore, one obtains $\partial_{t}F_{x_{1}t}=-i\omega
F_{x_{1}t}$ , $\partial_{t}j^{x_{1}}=-i\omega j^{x_{1}}.$ By utilizing Eqs.
(20), (21), (22) and taking the momentum limit, the Eq. (24) can be calculated
as

$\partial_{z}\sigma_{L}(\omega,z)=i\omega\Delta_{L}(\sigma_{L}(\omega,z)^{2}-(\Sigma_{L})^{2}),$
(26)

where

$\Delta_{L}=\frac{e^{\phi(z)}z(l^{2}\Omega^{2}f(z)-1)}{(l^{2}\Omega^{2}-1)f(z)},$
(27)

and

$\Sigma_{L}=\frac{e^{-\phi(z)}}{z}\sqrt{\frac{l^{2}\Omega^{2}-1}{l^{2}\Omega^{2}f(z)-1}}.$
(28)

The dynamical equation for the transverse channel is given by

$\displaystyle\partial_{z}j^{x_{2}}$
$\displaystyle-\frac{\sqrt{-g}}{h(\phi)}\Big{[}g^{tx_{1}}g^{x_{2}x_{2}}\partial_{t}F_{x_{1}x_{2}}-g^{tt}g^{x_{2}x_{2}}\partial_{t}F_{tx_{2}}$
(29)
$\displaystyle+g^{x_{1}t}g^{x_{2}x_{2}}\partial_{x_{1}}F_{tx_{2}}+g^{x_{1}x_{1}}g^{x_{2}x_{2}}\partial_{x_{1}}F_{x_{1}x_{2}}\Big{]}=0$
$\frac{h(\phi)g_{x_{2}x_{2}}g_{zz}}{\sqrt{-g}}\partial_{t}j^{x_{2}}+\partial_{z}F_{tx_{2}}=0,$
(30) $\partial_{x_{1}}F_{tx_{2}}+\partial_{t}F_{x_{2}x_{1}}=0.$ (31)

The transverse ”conductivity” and its derivative are defined as

$\sigma_{T}(\omega,z)=\frac{j^{x_{2}}(\omega,\vec{p},z)}{F_{x_{2}t}(\omega,\vec{p},z)},$
(32)
$\partial_{z}\sigma_{T}(\omega,z)=\frac{\partial_{z}j^{x_{2}}}{F_{x_{2}t}}-\frac{j^{x_{2}}}{F_{x_{2}t}^{2}}\partial_{z}F_{x_{2}t}.$
(33)

Similarly, we have $\partial_{t}F_{x_{1}x_{2}}=-i\omega
F_{x_{1}x_{2}},\partial_{t}F_{tx_{2}}=-i\omega
F_{tx_{2}},\partial_{t}j^{x_{2}}=-i\omega j^{x_{2}}.$ Therefore, the
transverse flow of Eq. (33) can be written as:

$\partial_{z}\sigma_{T}(\omega,z)=i\omega\Delta_{T}\big{(}\sigma_{T}(\omega,z)^{2}-(\Sigma_{T})^{2}\big{)},$
(34)

where

$\Delta_{T}=\frac{e^{\phi(z)}z}{f(z)},$ (35)

and

$\Sigma_{T}=\frac{e^{-\phi(z)}}{z}\sqrt{\frac{l^{2}\Omega^{2}f(z)-1}{l^{2}\Omega^{2}-1}}.$
(36)

By requiring the regularity of the horizon, one can obtain the initial
conditions for solving the equation $\partial_{z}\sigma_{L(T)}=0$. The flow
Eqs. (26) and (34) have the same form under a rotation of $\Omega$ = 0 GeV.
The spectral function can be defined by the retarded Green’s function as

$\rho(\omega)\equiv-\textrm{Im}G_{R}(\omega)=\omega\textrm{Re}\sigma(\omega,0).$
(37)

##  IV Numerical results of spectral functions in a rotating plasma

In this paper, we describe the spectral functions of heavy vector mesons
$(J/\psi$ and $\Upsilon(1S))$ of eq. (37) by computing Eq.s (26) and (34) and
utilizing the boundary conditions described in the previous section. The
spectral functions of heavy vector mesons $(J/\psi$ and $\Upsilon(1S))$ with
rotation are calculated by using the membrane paradigm Iqbal and Liu (2009) .

###  IV.1 The Result For $J/\psi$

Figure 1:  Spectral functions of $J/\psi$ at $T$ = 0.2 GeV , under different
chemical potentials $\mu$ and rotational radius $l$ . Figs. 1(a) and 1(b) are
parallel to the rotational angular velocity, while Figs. 1 (c) and 1(d) are
perpendicular to the rotational angular velocity.

In holographic rotating systems, the spectral function and the effective mass
should also depend on finite dimensions. Due to cylindrical symmetry, these
quantities depend on the radius $l$ mentioned in Sect. II . It would be
interesting to explore how various thermodynamic quantities of strongly
interacting rotating matter vary with the radius of the rotating system. The
nature of the rotating system as a function of the radius may be relevant to
future experimental observations. In addition, the radius is expected to
significantly change the angular momentum and moment of inertia. Therefore, it
becomes important to investigate how the spectral function and effective mass
of QCD matter under rotation depend on the rotational radius of the system.
Due to the limitation of the speed of light, it naturally leads to the
restriction $\Omega l\leq 1$.

Figure 1 presents the spectral functions of the $J/\psi$ vector mesons at
fixed rotational angular velocity $\Omega$, with different chemical potentials
$\mu$ and rotational radius $l$ . This clearly demonstrates the impacts of
density effects and rotational radius effects on the dissociation process.
There is a distinct peak corresponding to the $J/\psi$ meson state. The bell-
shaped curve represents the resonance state, and the peak position corresponds
to the resonance mass of the effective mass. Figures 1(a) and 1(b) exhibit the
spectral function parallel to the rotational angular velocity, while Figs.
1(c) and 1(d) exhibit the spectral function perpendicular to the rotational
angular velocity. We can see that the resonance peaks in Figs. 1(c) and 1(d)
are slightly lower than those in Figs. 1(a) and 1(b), especially at large
chemical potentials and large rotational radii, which means that the degree of
dissociation perpendicular to the rotational angular velocity direction is
greater than that parallel to the rotational angular velocity direction. As
both the chemical potential $\mu$ and the rotational radius $l$ increase, we
observe that the peak decreases, broadens, and slightly shifts to the right.
The study indicates that as the chemical potential and rotational radius
increase, they promote the dissociation of bound states and increase the decay
width and effective mass.

Figure 2:  Spectral function of $J/\psi$ for different rotation radii with
parallel and perpendicular to the rotational angular velocity at $\mu$ = 0.3
GeV , $\Omega$ = 0.5 GeV and $T$ = 0.2 GeV .

Figure 2 shows the spectral function of the $J/\psi$ at different rotation
radii. The left figure represents the longitudinal spectral function parallel
to the rotational angular velocity, and the right figure represents the
transverse spectral function perpendicular to the rotational angular velocity.
The resonance peak of $J/\psi$ decreases with increasing rotation radius for
$T$ = 0.2 GeV , $\mu$ = 0.3 GeV and $\Omega$ = 0.5 GeV. In addition, the
dissociation effect in the transverse direction is more pronounced than in the
longitudinal direction, and this effect becomes more prominent at larger
rotation radii. Therefore, it can be concluded that increasing the rotation
radius promotes the dissociation of bound states, and the dissociation
perpendicular to the direction of rotational angular velocity is more
significant than that parallel to it.

Figure 3:  The dependence of the effective mass on temperature at $\mu$ = 0.3
GeV and $\Omega$ = 0.5 GeV under different rotation radius.

Figure 3 shows the dependence of the effective mass of the peak position of
the $J/\psi$ spectral function on temperature at different rotation radii for
$\mu$ = 0.3 GeV and $\Omega$ = 0.5 GeV. The first interesting finding is that
the effective mass shows a slightly downward trend at lower temperatures, our
explanation for this phenomenon is as follows: at lower temperatures, quarks
are usually in a lower energy state. As the temperature increases, thermal
excitation causes quarks to excite to higher energy states, resulting in a
rearrangement of internal energy levels. Thermal excitation can reduce the
effective potential of bound states, resulting in a decrease in the effective
mass of mesons.

At higher temperatures, the influence of temperature on the system is more
significant. The increase in temperature leads to an increase in the distance
between quark-antiquark pairs, thereby enhancing the interaction between
quark-antiquark and the medium. The second interesting finding is that in both
the transverse and longitudinal directions, in the lower temperature region
$(T\leq$ 0.42 GeV), the effective mass slightly increases with the increase of
the rotation radius, while in the higher temperature region $(T>$ 0.42 GeV),
the effective mass slightly decreases with the increase of the rotation
radius. However, for the transverse rotational direction, when the rotation
radius reaches $l$ = 1.2 GeV-1 , the effective mass has a maximum near $T$ =
0.65 GeV , and then decreases sharply with increasing temperature. Combining
with Fig. 2, one finds that when the rotation radius reaches $l$ = 1.2 GeV-1 ,
the peak of the transverse spectrum is very low, indicating $J/\psi$
dissociation. This indicates that the dissociation radius of the transverse
spectrum of the $J/\psi$ is smaller than that of the longitudinal spectrum.

###  IV.2 The Result For $\Upsilon(1S)$

Figure 4:  Spectral functions of the $\Upsilon(1S)$ at $T=0.265$ GeV, under
different chemical potentials $\mu$ and rotational radius $l$. (a) and (b) are
parallel to the direction of rotation, while (c) and (d) are perpendicular to
the direction of rotation.  Figure 5:  Spectral functions of the
$\Upsilon(1S)$ for different rotation radius. At $\mu$ = 0.3 GeV and $\Omega$
= 0.5 GeV, it shows the spectral functions of the $\Upsilon(1S)$ states for
different rotational radius at $T$ = 0.265 GeV.  Figure 6:  The dependence of
effective mass on temperature at $\mu$ = 0.5 GeV and $\Omega$ = 0.75 GeV,
under different rotation radius.

Figures 4, and 5 show the spectral functions of longitudinal(L) and
transverse(T) of the $\Upsilon(1S)$ states under similar conditions. Compared
with the previous results for the $J/\psi$ state, it can be seen that the
difference lies in the influence of the rotation radius on the peak position
of the $\Upsilon(1S)$ spectral function. These observations indicate that the
increasing rotation radius in the rotating system can promote the dissociation
process.

As shown in Fig. 6, both the longitudinal and transverse effective masses
increase with increasing temperature, which is consistent with the calculation
results of the $J/\psi$. When the rotation radius is small, the influence of
the rotation radius on the relationship between effective mass and temperature
is not significant in both longitudinal and transverse directions. However, as
the rotation radius increases, the influence of the rotational radius on the
relationship between effective mass and temperature becomes significant. The
transverse effective mass varying with temperature shown in Fig. 6(b) exhibits
a similar phenomenon to that of the transverse effective mass of the $J/\psi$,
reaching a maximum value when the temperature reaches a certain value ($T$ = 1
GeV).

##  V SUMMARY AND CONCLUSIONS

In this paper, we focus on the impact of rotation on the spectral function of
the $J/\psi$ and the $\Upsilon(1S)$ states. The metric by cylindrical
coordinates with rotation is introduced into the system to calculate the
spectral functions of heavy vector mesons. By incorporating rotation medium
into the holographic QCD, we investigate the influence of rotational effect on
the properties of the $J/\psi$ and the $\Upsilon(1S)$ . From the results of
this study, it is clear that temperature, chemical potential, and rotational
radius effects enhance the dissociation process of heavy vector mesons within
the medium. This rotation-induced effect is more significant for heavy vector
mesons in the transverse direction.

Figure 7:  The Spectral functions of the $J/\psi$ and the $\Upsilon(1S)$
states for different temperatures $T$ at $\mu$ = 0.3 GeV , $\Omega$ = 0.5 GeV
and $l$ = 1.0 GeV-1 . (a) and (b) are for the $J/\psi$ states, and (c) and (d)
are for the $\Upsilon(1S)$ states; (a) and (c) are for the longitudinal
direction, while (b) and (d) are for the transverse direction

In this article, we mainly study the influence of the rotation radius on the
spectral function. Here we need to add a discussion on the relationship
between the spectral function and temperature, as shown in Fig.7. The spectral
functions of the $J/\psi$ and the $\Upsilon(1S)$ states with three different
temperature values at $\mu$ = 0.3 GeV, $\Omega$ = 0.5 GeV and $l$ = 1.0 GeV-1
are presented in Fig.7. This clearly demonstrates the impact of thermal
effects on the dissociation process. It is found that as the temperature
rises, the peak decreases and widens, and shifts to the right. This indicates
that as the temperature increases, it promotes the dissociation of bound
states and increases the decay width and effective mass.

The first holographic study on the influence of the radius of a homogeneous
rotating system on the spectra functions for heavy vector mesons is
established in the article. As we are discussing the rotating system of QCD
medium, the spectra functions characteristics should depend on the finite size
of the rotating system. Due to the cylindrical symmetry of the rotating
system, the rotation radius $l$ has become an important characteristic
quantity of the rotating system. Studying the dependence of phase transition
characteristics of the strongly interacting rotating matter on the radius of
the rotating system is an important research topic. It is found that
increasing the rotation radius promotes the dissociation of the $J/\psi$ and
the $\Upsilon(1S)$ states, and the dissociation perpendicular to the direction
of rotational angular velocity is more significant than that parallel to it at
large rational radius.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Grants No. 11875178, No. 11475068, No. 11747115).

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