11institutetext: University of Applied Sciences and Arts Northwestern
Switzerland (FHNW), 5200 Windisch, Switzerland
11email<EMAIL_ADDRESS>22institutetext: Astronomy and Astrophysics
Section, School of Cosmic Physics, Dublin Institute of Advanced Studies, 31
Fitzwilliam Place Dublin 2, D02XF86, Ireland 33institutetext: University of
Graz, UniversitÃtspl. 3, 8010 Graz, Austria 44institutetext: Department of
Physics & Astronomy, Western Kentucky University, Bowling Green, KY 42101, USA
55institutetext: ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
66institutetext: Ateleris, Badenerstrasse 13, CH-5200 Brugg Switzerland
77institutetext: Leibniz-Institut fÃr Astrophysik Potsdam (AIP), An der
Sternwarte 16, D-14482 Potsdam, Germany 88institutetext: UniversitÃ Paris-
Saclay, UniversitÃ Paris CitÃ, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France
99institutetext: MIDA, Dipartimento di Matematica, UniversitÃ di Genova, via
Dodecaneso 35, I-16146 Genova, Italy 1010institutetext: Istituto Nazionale di
Astrofisica, Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025
Pino Torinese, Italy

# The data center for the Spectrometer and Telescope for Imaging X-rays (STIX)
onboard Solar Orbiter

Hualin Xiao 11 Shane Maloney 22 Säm Krucker 11 Ewan Dickson 33 Paolo Massa 44
Erica Lastufka 11 Andrea Francesco Battaglia 1155 LÃ¡szlÃ Etesi 11 Nicky
Hochmuth 66 Frédéric Schuller 77 Daniel F. Ryan 11 Olivier Limousin 88 Hannah
Collier 1155 Alexander Warmuth 77 Michele Piana 991010

(September 3, 2024)

###### Abstract

The Spectrometer and Telescope for Imaging X-rays (STIX) on board Solar
Orbiter observes solar X-ray emission in the range of 4 – 150 keV and produces
spectra and images of solar flares over a wide range of flare magnitudes.
During nominal operation, STIX continuously generates data. A constant data
flow requires fully automated data-processing pipelines to process and analyze
the data, and a data platform to manage, visualize, and distribute the data
products to the scientific community.

###### keywords:

Solar flares – Data platform – STIX data products – X-ray imaging – Data
processing pipeline

The STIX Data Center has been built to fulfill these needs. In this paper, we
outline its main components to help the community better understand the tools
and data it provides. The STIX Data Center is operated at the University of
Applied Sciences and Arts Northwestern Switzerland (FHNW) and consists of
automated processing pipelines and a data platform. The pipelines process STIX
telemetry data, perform common analysis tasks, and generate data products at
different processing levels. They have been designed to operate fully
automatically with minimal human intervention. The data platform provides web-
based user interfaces and application programmable interfaces for searching
and downloading STIX data products. The STIX Data Center has been operating
successfully for more than two years. The platform facilitates instrument
operations and provides vital support to STIX data users.

##  1 Introduction

Solar Orbiter is an ESA-led mission with a suite of ten remote-sensing and in
situ instruments to study the interaction between the Sun and the heliosphere.
Launched on 10 February 2020 into a heliocentric orbit, its nominal mission
will last seven years with a planned extension of three additional years
(Müller et al., 2020). Solar Orbiter’s closest approach to the Sun is 0.28 AU
and its orbit will shift out of the ecliptic as the mission progresses,
providing better views of the solar poles (Forveille & Shore, 2020).

Solar Orbiter’s Spectrometer Telescope for Imaging X-rays (STIX; Krucker et
al., 2020) observes X-rays from 4 to 150 keV. Its main scientific objective is
to study the hot solar plasma and accelerated electrons produced during solar
flares. STIX consists of 32 subcollimators, each with a pair of slightly
offset grids that encode spatial information in a MoirÃ pattern which is
detected by a pixelated cadmium telluride (CdTe) detector. Images with an
angular resolution of a few arcseconds can be produced on the ground with
Fourier-based image reconstruction techniques. STIX’s detectors are
spectroscopic and so they enable the thermal plasma and accelerated electrons
to be characterized via the spectra they produce. They also enable images to
be produced in any energy range between 4 to 150 keV and over any integration
time greater than $\sim$0.5 s. This allows scientists to customize the imaging
based on their science goals and observational constraints. Hence, STIX
provides observations of the intensity, energy, dynamics, and location of
accelerated electrons and heated plasma during solar flares. For more
information on STIX instrumentation and its scientific capabilities, readers
can refer to Krucker et al. (2020).

During normal operations, STIX observes continuously. HouseKeeping (HK) and
QuickLook (QL) data are generated automatically and sent to Earth at regular
intervals. Science-quality data products are generated on board only in
response to requests from the ground. These products are clipped and summed in
time, energy, and pixels depending on the quality of the observation and then
further compressed to optimize the telemetry needs on the instrument. This
results in hundreds of different types of raw telemetry data packets. To
handle these, the STIX Data Center has developed a data platform and automated
data-processing pipelines which assist with instrument operations, simplify
STIX data analysis, and increase data accessibility for the solar physics
community. The pipelines convert raw telemetry data into QLs and data products
at different processing levels which can be used for scientific analysis. The
platform stores all STIX data products and provides web-based tools for users
to explore and analyze STIX data.

In this paper, we describe the processing pipelines, core analysis algorithms,
data products, and other tools used and provided by the STIX Data Center. The
aim is to help users better understand the data center and hence how to best
utilize it and the STIX instrument.

##  2 STIX telemetry data products

STIX generates many different types of raw telemetry packets. However, only
three categories are relevant to scientific users: HK, QL, and science data.
HK and QL data are directed to the low-latency data stored in the spacecraft’s
solid-state mass memory (SSMM) and sent to the ground with the highest
priority. Except for the raw pixel data product, all count- and trigger-based
values in QL and science data products are compressed with an integer
compression algorithm (Krucker et al., 2020).

###  2.1 HK data

The HK data monitor the instrument’s status and performance in order to ensure
it functions properly. STIX generates HK packets continuously while being
powered. The HK packets include details such as the instrument’s temperature,
voltages, currents, CPU usage, the status of the attenuator switches, trigger
rates, file system information, and aspect system readouts (Warmuth et al.,
2020; Krucker et al., 2020). STIX typically generates a HK packet every 64
seconds.

###  2.2 QL data

The QL data are only generated when STIX is in the nominal observation mode,
as they require the detectors to be powered and operating. There are five
types of QL data:

  * •

QL light curves contain time series of 4-second-integrated, detector-summed
counts in five energy bands. Furthermore, they include the corresponding
detector-summed triggers, and the rate control regime state (i.e., attenutator
state; see Krucker et al., 2020). We note that the QL light curves do not
include time series from STIX’s two special detectors, the BacKGround monitor
(BKG) and the coarse flare locator (CFL). It is worth mentioning that STIX
does not correct for detector dead time, absorption of X-rays by the entrance
and grids, impacts of the presence of the attenuator, or rate control regime
states on board. Hence the effects of these phenomena are present in the QL
light curves.

  * •

QL background light curves: STIX uses the background monitor – a detector with
a special grid aperture – to monitor both the X-ray background and the intense
unattenuated X-ray fluxes from large solar flares. It consists of an open
front grid window and a rear grid window that is fully opaque except for six
small openings. QuickLook background light curves contain counts and triggers
of all the detector pixels integrated over 8 seconds in the same five energy
bands as the QL light curves.

  * •

QL variance data are the onboard computed variance of 40 successive detector-
summed count rates based on a 0.1-second integration.

  * •

QL spectra are produced for each detector by integrating over all its pixels
and accumulated over 32 seconds. During nominal operations, a QL spectrum for
each detector is generated every 1024 seconds.

  * •

Calibration spectra are high energy resolution, per pixel, energy spectra in
ADC units (ADU) accumulated for photons emitted by 133Ba radioactive sources
located within the STIX instrument. STIX generates a calibration spectrum for
each pixel every 24 hours during normal operations. These are used to
calibrate the gain of each pixel on the ground by observing the locations of
the well-known spectral line peaks. If the gain is seen to shift, a new ADU to
keV conversion table is uploaded for each pixel from the ground.

###  2.3 Science data

Science data products are only generated and down-linked in response to a data
request from the ground. They do not present a continuous record of solar
activity because of the limited bandwidth and onboard storage. Instead, they
cover scientifically interesting periods identified on the ground from the
low-telemetry QL data. Science data are generated from the pixel data stored
in the STIX onboard archive memory. They are rebinned in time, energy, pixels,
and detectors in accordance with the parameters of the data request. These
parameters are chosen on the ground on a case-by-case basis to optimize both
telemetry efficiency and scientific return. Due to the limited storage
available on board, raw pixel data are overwritten after a few months.
Therefore, bulk science data (BSD) requests must be made within the following
weeks of observations being made.

STIX can generate six different types of science data products:

  * •

Raw pixel data are the least processed data product, which provides time-
binned pixel counts in the highest available time and energy resolution,
generally with a dynamic accumulation time between 0.5 s and 20 s depending on
solar activity. The raw pixel values are not integer-compressed, and hence
require a large amount of telemetry. Therefore, they are rarely requested and
are used primarily for testing and verification.

  * •

Compressed pixel data are based on the raw pixel data product. They combine
the highly resolved pixel counts into larger user-defined time and energy
bins. In addition to the rebinning, they are integer-compressed on board
before being sent to the ground. This is the most common BSD product and is
well suited for scientific analysis.

  * •

Summed and compressed pixel data are also based on the raw pixel data product.
Similar to the compressed pixel data, they combine the raw pixel counts into
larger user-defined time and energy bins, but they also sum the counts over
multiple pixels. The pixel sums can be configured by telecommand. By default,
the top and bottom pixels in the same column are summed, reducing the apparent
number of pixels per detector to four(Krucker et al., 2020). Therefore, it is
a suitable option when telemetry is limited. In addition, the summed pixel
data values are integer-compressed.

  * •

Visibility data: Each STIX subcollimator measures a spatial Fourier component
on the sky which can be combined into images on the ground using indirect
imaging algorithms. Each Fourier component is encoded into an intensity sine
wave which is sampled at four locations by the subcollimator’s pixel columns.
This information can be further encoded into a complex number which gives the
amplitude and phase of the sine wave (Massa et al., 2022). This format, known
as visibility, is twice as data efficient as the summed pixel column values.
Because STIX has been given a larger telemetry budget than originally
anticipated, this data product is rarely requested because the visibility
calculation can be done slightly more accurately on the ground. However,
should telemetry be significantly reduced in the future, this product can be
requested with a limited loss of scientific value.

  * •

Spectrogram data are summed over selected pixels, detectors, and time-energy
bins. The pixel and detector summing does not lead to a significant loss of
spectral resolution, but it does eliminate the spatial information. In almost
all cases, however, spectrogram data are equally useful for spectral analysis
as compressed pixel data, but they require two orders of magnitude less
telemetry. This means that spectral analysis can be performed at high
resolution and time cadence for longer time periods than would otherwise be
possible. Hence, along with compressed pixel data, spectrogram data are the
most requested science data product. It is worth mentioning that the
spectrogram data are available at the highest resolution possible at all times
that STIX is in the nominal observation mode since January 2022.

  * •

High time-resolution aspect data: STIX stores readouts from the four
photodiodes in the aspect system in the onboard archive memory at a resolution
of 16 ms. So-called burst-mode aspect data can be later requested at full
resolution, or binned to user-specified resolution. This is especially useful
to accurately measure the direction where STIX is pointing, even when the
spacecraft attitude is changing rapidly.

Science data are transmitted to the ground in raw binary packets. These are
labeled level 0 at the STIX Data Center.

##  3 Data flow

Figure 1:  STIX Data Center hardware architecture and data flow. Solar Orbiter
telemetry data are processed at the Mission Operations Center (MOC) and
delivered to the instrument team via the generic file transfer system (GFTS).
The STIX Data Center currently utilizes five servers: one for data exchange,
two for running the data-processing pipelines, one for the web server, and one
for data storage (network attached storage or NAS). STIX data products are
stored on the NAS server. The data exchange server is also used to deliver
instrument operations requests (IORs) to the MOC, and payload direct operation
requests (PDORs) to the Science Operations Center (SOC).

During nominal science operations, low-latency data are transmitted to Earth
during every ground station pass regardless of orbital geometry, whereas
science data are only downlinked when the bandwidth permits. Every instrument
team is responsible for keeping the transmitted data volume within
preallocated boundaries.

Telemetry data received by ground stations are processed by the Mission
Control System at the Solar Orbiter Mission Operations Center (MOC). Telemetry
data after undergoing quality checks, packet extraction, and filling are
stored in the EGOS Data Dissemination System (EDDS) (Peccia, 2005a, b). They
are distributed to the instrument teams regularly via the generic file
transfer system (GFTS). As shown in Fig. 1, the STIX Data Center uses a
dedicated server to exchange data between the STIX Data Center and ESA. In
addition to receiving telemetry, it is also used to deliver operation requests
to the Science Operations Center (SOC) or the MOC. Telemetry files received by
the data exchange server are transferred to network attached storage (NAS),
and are subsequently processed by the pipelines introduced in the next
chapter.

Low-latency telemetry data generally arrive at the STIX Data Center within a
few days, depending on the next ground station pass. However, science data may
be delayed several weeks after being generated on board due to possibly low
bandwidth or priority. In addition to telemetry data, the STIX Data Center
receives auxiliary data (Acton, 1996; Acton et al., 2018) from the SOC, which
contain information on spacecraft ephemeris, attitude, and calibration factors
required for the conversion of onboard timestamps to UTC times.

##  4 Data-processing pipelines

###  4.1 Data-processing pipeline overview

Figure 2: Telemetry processing pipelines at the STIX Data Center. Raw packets
are processed to level 1, which includes parsing, integer decompressing, and
timestamp conversion. Level-1 packets are recorded in a NoSQL database. Then
they are selected and processed in four different paths, each of which is
responsible for processing specific data types.

STIX telemetry data are processed immediately upon reception at the STIX Data
Center by the data-processing pipeline shown in Fig. 2. Raw packets are
processed to level 1 (L1) which includes parsing, integer decompressing, and
timestamp conversion. Level-1 packets are stored in a tree-like structure and
recorded in a NoSQL database. Then they are selected and processed in four
different paths. In the first, the HK, QL, and science data are used to create
L1 FITS files. The second path computes energy calibration factors from
calibration spectra. In the third path, solar flares are identified by
applying a detection algorithm to QL data. The fourth path performs common
analyses of science data.

###  4.2 Raw packet parsing

Raw telemetry data at the STIX Data Center are stored as binary packets. Each
packet contains a fixed-length header and a list of parameters that vary with
the type of packet. The parsing of parameters is based on information in the
Mission Information Base (MIB), which contains the name and length of each
parameter for each type of packet. The parsed packets contain raw values of
parameters, which need to be converted to physical values. Raw values of
spacecraft-local times are converted to UTC times with the latest version of
SPICE kernels (Acton, 1996; Acton et al., 2018; and, 2019). Raw values of HK
parameters are converted to physical values using the ground-calibrated
conversion factors stored in the MIB. Compressed counts in science data are
decompressed using a lookup table. After the above processing steps, packets
are organized in a tree-like structure. They are considered L1 packets and
recorded in a collection in the NoSQL database. The NoSQL database is
schemaless, which means that the data format of each record can be different
and a predefinition of the data format is not required. This makes it very
suitable for storing the tree-like structure L1 data packets. The NoSQL
database is extremely convenient for tasks such as searching, sorting,
organizing, merging data packets, and verifying data integrity.

In addition to L1 packets, other metadata, such as filenames, SPICE kernel
version, and the MIB version, are recorded in another collection in the NoSQL
database. This allows for a rapid query of the associated raw telemetry data.

####  4.2.1 FITS products

The flexible image transport system (FITS) is a portable file standard widely
used in astronomy to store, transmit, and manipulate scientific images,
tables, and associated data (Pence, W. D. et al., 2010). Therefore, the FITS
format is adopted by the STIX Data Center to store the standard data products.
After parsing each new raw telemetry file, HK, QL, and science packets are
sequentially selected from the NoSQL database and merged after passing checks
for data integrity and consistency. The merged data as well as the associated
metadata and auxiliary data are written to FITS files. In the meantime, their
metadata are recorded in a collection in NoSQL, which allows for the fast
query of the products. The FITS files, created from L1 packets immediately
upon the parsing process, are defined as level-1A (prereleased L1) products.
They are used by some subsequent data-processing pipelines.

FITS files are recreated in a similar pipeline after a few days to weeks after
all inputs are validated manually. The created FITS files are regarded as the
formal L1 products at the STIX Data Center. In most cases, the FITS files at
level-1A and L1 are almost identical, except that the level-1A FITS files may
use predicted ephemeris data. In addition, level-1A FITS files may not contain
all header keywords required by the ESA standard. As such, L1 FITS files are
recommended for STIX data users whenever they are available.

###  4.3 Energy calibration

Figure 3: Example of STIX in-flight calibration spectrum recorded by a single
pixel. The most prominent peaks, from left to right, are the 31, 35, and 81
keV photopeaks. The first two are fitted with a double-Gaussian function, and
the high energy peak with a crystal-ball function.

STIX converts ADC channels to "science energy channels" on board by the FPGA
using an energy lookup table (ELUT), which defines the ADC channel edges for
each science channel for each pixel. An ELUT can be constructed using energy
conversion factors determined from calibration runs. STIX continuously
accumulates an energy spectrum (in ADC units) for each pixel separately for
events from the onboard Ba133 sources and formats a spectrum typically every
24 hours. Fig. 3 shows an example of such a spectrum. The three most prominent
peaks are produced by photons of 31 keV, 35 keV, and 81 keV energy from the
calibration sources. To determine the positions of the photopeaks, the first
two peaks are fitted with a double-Gaussian function, and the third peak with
a crystal-ball function (Skwarnicki, 1986), which consists of a Gaussian core
and a power-law low-end tail, below a certain threshold. Then a linear fit is
made to the positions and keV energies whose slope and intercept give the gain
(i.e., the ADC to energy conversion factor) and baseline, respectively. The
ECC method (see Maier & Limousin, 2016; Maier et al., 2020) is another method
often used by the STIX team to determine the calibration factors. We found
that the results of the two methods are consistent within 1$\sigma$.

The above steps are performed for each calibration spectrum once the data are
available at the STIX Data Center. The calibration factors are recorded in a
collection in the NoSQL database and used for further correction of energy
bins in offline data analysis. Once significant changes in the calibration
factors are observed, the STIX operations team creates a new ELUT and uploads
it to STIX.

###  4.4 Solar flare identification

STIX identifies solar flares on board based on detector count rates, and the
results are included in the QL data (Krucker et al., 2020). However, the data
only provide limited information on flares due to the constraints of the
telemetry budget and onboard computing resources. Moreover, microflares are
not reported due to the relatively high trigger threshold. It is necessary to
maintain a flare list on the ground for requesting science data and also for
users to find events of interest.

Figure 4: STIX 4 – 10 keV QL light curve for 2022-01-02T00:30:00 –
2022-01-02T09:30:00 (UTC) and identified flares. The orange curve is the
smoothed light curve using a moving average filter with a time window of 1
minute. The green curve is the estimated signal envelope using the SNIP
algorithm. The identified peaks are marked with plus signs, and the gray
ranges show their time ranges.

Using QL light curves, solar flares can be identified in greater detail on the
ground. The ground identification procedure includes the following steps:

  * •

Light curve smoothing: The selected light curve is filtered using an
unweighted moving average filter with a time window of 1 min. This can smooth
out statistical fluctuations and electric surge spikes.

  * •

Envelope subtraction: A flare may last hours, and there may be short-duration
pulses lying on the envelope (the main pulse) in the light curve. To
facilitate the identification of those short-duration pulses, the envelope is
subtracted from the smoothed light curve, which is estimated using the SNIP
algorithm (Ryan et al., 1988).

  * •

Identification of flare peaks: We consider that a flare is detected if the
peak count rate after envelope subtraction lies beyond two standard deviations
of the mean count rate during quiet Sun periods. The flare start and stop
times are given at the times exceeding this threshold.

  * •

Merging of flare peaks: Two peaks are considered from one single flare if the
peak times differ by less than 5 min. This can reduce the number of reported
flares and simplify data analysis. We note that this can also affect any
statistical flare distribution derived from this list.

As an example, Fig. 4 shows the STIX QL light curve in the energy range 4 – 10
keV, recorded from 2022-01-02T00:30:00 to 2022-01-02T09:30:00 (UTC). The
orange curve is the smoothed light curve. The identified peaks are marked with
the plus signs, and the colored ranges show the time ranges.

The above steps are repeated for QL light curves of the other four higher
energies, which provide information on the upper limit of the X-ray energy of
the flare. The time ranges, peak count rate, and total counts as well as the
corresponding ephemeris data of the identified flares are stored in a
collection called the flare list in the NoSQL database, which is later used
for the creation of data requests (see 5).

###  4.5 Solar flare standard analysis pipeline

####  4.5.1 Estimation of solar flare GOES class

Figure 5: Scatter plot of GOES/XRS low channel peak flux with respect to STIX
4 – 10 keV count rates scaled to 1 AU for 717 solar flares observed by both
GOES and STIX during the cruise phase. The solid line is a linear fit to the
log-log graph. The Pearsonâs correlation coefficient in the log-log scale is
0.91. From the fit, we get the following GOES flux estimation formula:
$f=10^{-7.376+0.622\log_{10}(X^{{}^{\prime}})}$ (in units of W/m2), where
$X^{{}^{\prime}}$ is the STIX peak count rate corrected for the distance
variations between the Sun and Solar Orbiter.

Solar Orbiter is often far from the Sun-Earth line and hence a considerable
number of flares observed by STIX are not observed by GOES satellites (and
vice versa). In order to estimate the GOES classes of such flares, we selected
717 solar flares observed by both GOES/XRS and STIX. Fig. 5 shows the scatter
plot of the peak fluxes measured by GOES satellites with respect to the STIX
background-subtracted count rates at the peaks, in the energy range of 4 to 10
keV. STIX count rates $x$ have been corrected for the different distances
between Solar Orbiter and the Sun using $X^{{}^{\prime}}=xr^{2}$, where $x$ is
the count rate after background subtraction and $r$ is the distance between
the Sun and Solar Orbiter in units of AU. A clear correlation (The correlation
coefficient $r=0.90$) is seen in Fig. 5. The wide spread at low fluxes can be
explained by the difference in the energy response of the two instruments and
the variation in flare temperatures. The correlation can be fitted with a
linear fit in the log-log scale. From the fit, we get the following GOES flux
estimation formula: $f=10^{-7.376+0.622\log_{10}(X^{{}^{\prime}})}$ (in units
of W/m2), where $X^{{}^{\prime}}$ is the STIX peak count rate corrected for
the distance variations between the Sun and Solar Orbiter. It is currently
used to estimate the GOES classes of flares that are not directly observed by
GOES satellites. The estimated GOES fluxes are stored in the flare list
collection in the database.

####  4.5.2 Estimation of coarse flare locations using CFL data

STIX estimates approximate flare locations on board by maximizing the
correlation between observed counts recorded by the 12 pixels in its coarse
flare locator (CFL) subcollimator with expected counts using a lookup table
(Krucker et al., 2020). With the requested science data, coarse flare
locations can be reconstructed more accurately, as it allows for more
sophisticated algorithms, as well as greater flexibility in selecting a time
and energy range to be integrated.

When the pixel data of a flare are available at the STIX Data Center, its
flare location is estimated. The steps are as follows:

  1. 1.

Counts are integrated around the peak for each CFL pixel.

  2. 2.

The background is subtracted using a background file.

  3. 3.

The illuminated area is estimated on each CFL pixel. This is based on two
assumptions: the illuminated area of a pixel is proportional to its relative
count rate, and the total illuminated area of the imaging detectors is
independent of the source location.

  4. 4.

The flare location is estimated by minimizing the weighted sum of squared
deviations (i.e., weighted chi-squares) between the calculated illuminated
areas and expectations simulated for potential flare locations in a 400
$\times$ 400 grid.

Figure 6:  Example of a coarse flare location solution. Left: Calculated areas
of illuminated regions of the 12 CFL pixels and the best-fit simulated pattern
for the flare location at (550, 278) arcsec. Pixels 0 to 3 are the top big
pixels, pixels 4 to 7 are the bottom pixels, and pixels 8 to 11 are the small
pixels as shown in the right panel. Middle: Best-fit flare centroid location
(marked by x) and its 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence contours.
Right: Projection of CFL subcollimator (the gray shaded regions) on the
detector pixels (the green closed areas) simulated for the best-fit flare
location. The top row, bottom row, and small pixels (from left to right) are
numbered sequentially from 0 to 11.

As an example, the left panel of Fig. 6 shows the calculated and best-fit
illuminated areas of the CFL pixels for the flare observed at
2021-05-07T19:00:00 (GOES class M3.9); the middle panel shows the best-fit
flare centroid location, as well as its $1\sigma$, $2\sigma$, and $3\sigma$
contours. The simulated CFL shadow pattern is shown in the right panel. The
estimated flare locations are stored in the flare list in the database.

####  4.5.3 Imaging and spectroscopy pipeline

Figure 7:  STIX QL light curves and reconstructed images of the solar flare
observed at 2022-10-07T13:50:39 (UTC), created by the image reconstruction
pipeline. A 1-min integration time around the peak was selected and the images
were reconstructed using four different algorithms: Back-Projection (BP),
CLEAN, MEM_GE, and VIS_FWDFIT. The full-disk BP image shown in the top row of
the right panel has been used to identify the source location. The other
reconstructions were performed around the location of the flare. We note that
the multiple circular Gaussian shapes have been selected for reconstructing
the flaring source by means of VIS_FWDFIT (bottom-right panel).

To help identify flares of interest, we developed an imaging and spectroscopy
pipeline that automatically reconstructs images and performs spectral analysis
for each flare with detector summed counts greater than 10,000 after receiving
its pixel data. The pipeline first selects and integrates counts for 60
seconds around the peak of the flare for each pixel, and then subtracts the
background from the integrated counts using the pixel data acquired during a
quiet solar period. It is worth mentioning that quiet solar periods are
determined on the basis of counts in the QL light curves. Subsequently, the
transmission and dead-time corrections are performed on the background-
subtracted counts. Then the corrected counts are further converted into the
visibilities of two energy bands of 4 – 10 keV (thermal energy) and 16 – 28
keV (nonthermal energy). Then the visibilities are used for image
reconstruction using four different algorithms: Back-Projection (BP) (Massa et
al., 2022), CLEAN (Högbom, 1974), MEM_GE (Cornwell & Evans, 1985; Massa et
al., 2020), and VIS_FWDFIT (Volpara et al., 2022). Finally, the reconstructed
images are corrected for spacecraft off-pointing and rotations. As an example,
the first panel of Fig. 7 shows the light curves and time range selected for a
flare that occurred at 2022-10-07T13:50:39 (UTC). The rest of the panels show
the images reconstructed with the algorithms. In addition to image
reconstruction, the transmission and the dead-time corrected counts are used
for spectral analysis. Fig. 8 shows the results of the spectral fitting for
the same flare as in Fig. 7. The spectrum is fitted with a thermal component
and a nonthermal component.

The results of the pipeline are saved to files in both FITS and PNG formats.
At the same time, the indexing information from the file, the parameter values
of the spectral analysis, and the auxiliary data are recorded in a collection
in the database.

Figure 8: Spectral fitting results for the same flare as in Fig. 7. The
spectrum is fitted with a thermal component and a nonthermal component.

##  5 Science data request strategy

As mentioned previously, high-resolution pixel data are only downlinked in
response to requests from the ground. They are stored in the onboard archive
memory for a few months before being overwritten by new data. They are
processed and downlinked after receiving data request telecommands from the
ground. A data request telecommand contains information about the selected
data and values of parameters required to process the data on board, including
the data compression level, time range, minimal time bin, energy bin width,
and masks indicating detectors and pixels to be selected. STIX detects
thousands of flares per year; therefore, selecting data is a tedious task, as
many factors must be considered, such as count rates, time binning of data,
statistics of selected data, and also the telemetry budget. The data selection
strategy has been continuously optimized over the past two years. The current
strategy is as follows:

  * •

Compressed L1 pixel data are requested for each of the ground-identified
flares with a total number of signal counts in the QL light curves greater
than 10,000, which is approximately the minimum number of counts needed to
reconstruct an X-ray image. The requested energy range is chosen to be the
range in which obvious signal counts are seen, whereas, the requested time
resolution is adjusted based on the telemetry budget and the scientific
importance of the flare. If the peak count rate is above 125 counts/sec
(approximately equivalent to the count rate observed for a B3 flare at 1 au),
pixel counts with a high time resolution are requested. Otherwise, pixel
counts are integrated over the whole flaring time to reduce the telemetry data
volume. These parameters may be customized for individual flares by human
operators during review.

  * •

Spectrograms with the highest time and energy resolution are requested for all
periods when STIX is in its observation mode.

  * •

Time-integrated pixel data for background subtraction: Time-integrated pixel
data with durations of one or two detector temperature cycles (each cycle
lasts about 40 minutes), which are acquired during quiet-Sun periods, are
requested. The data are used for background subtraction when performing
spectroscopy and imaging.

  * •

High time-resolution aspect data are requested, for example, for periods when
the spacecraft’s attitude changes drastically. Such periods can be known from
the SPICE kernels or the aspect system readouts in HK data.

The selection of science data of the above types is done automatically using a
program (except for aspect data). The information of the selected data is
written into a collection in the NoSQL database. In addition, the STIX
operations team also selects data for special needs. After being checked and
adjusted by the STIX operations team, groups of new data requests that meet
the operations requirements are selected from the database and then compiled
into instrument operation requests (IORs), which are used to create final
telecommands at the MOC. These telecommands are then uploaded to the
spacecraft and executed by STIX typically two to three weeks later.

Figure 9:  Interactive web-based STIX QL data browser. In addition to STIX
QLs, it can also display QLs of simultaneous measurements performed by other
instruments, such as GOES/XRS, SDO/AIA, and EUI on board Solar Orbiter. The
browser is available at the link
%****␣data_center.tex␣Line␣425␣****https://datacenter.stix.i4ds.net/view/ql/lightcurves.
Figure 10:  Client side and server side data exchange mechanism.

##  6 STIX Data Center user interfaces

###  6.1 Interactive web pages

The data center platform provides various HTTP interfaces (APIs) that allow
access to STIX data products and the NoSQL database via HTTP requests. We have
built dozens of web applications to manage and browse STIX data based on these
APIs. Web techniques are chosen because they offer many advantages, such as
clear cross-platform usability, broad access through browsers, rapid
development, and easy maintenance.

As an example, Fig. 9 shows a screenshot of the STIX QL data browsing tool. It
allows users to browse available QL data interactively. It interacts with the
server through APIs. The data exchange mechanism between the server side and
the client side is shown in Fig. 10. After receiving a request from the client
side, the server retrieves QL counts from the NoSQL database for the user-
specified time range. After excluding duplicates and merging, the server sends
QL counts and metadata in JSON format back to the client. The data are then
used to create interactive light-curve plots using JavaScript on the client
side. The interactive plot uses state-of-the-art web technologies that enable
users to perform a range of operations, such as rebinning the integration
time, correcting the light travel time between the spacecraft and the Earth,
and exporting plot data to a local file. In addition, QLs from other solar-
observing instruments can also be displayed on the same page after users’
activation, making it easier to find events of interest for joint analysis.

Based on similar concepts, tens of web tools have been developed to browse
other STIX data products. The four most commonly used tools are listed below:

  * •

The science data manager and browser provides users with tools for searching,
downloading, visualizing, and analyzing science data. The interactive analysis
tools allow users to select data of interest for standard analysis tasks, such
as background subtraction and energy rebinning, without installing additional
software. The algorithms are implemented on the client side using JavaScript.
In addition, users can submit imaging and spectroscopy tasks to the server and
view the results on the same page. This reduces the barriers to exploring STIX
data for new users and is convenient for experienced users.

  * •

The preview images and spectroscopy product viewer is a web-based tool for
managing and viewing the imaging and spectroscopy results. The viewer also
provides tools for plotting the time evolution of emission measures and
temperatures, creating animations of X-ray images for the selected runs,
generating IDL or Python templates that allow for the same results to be
reproduced on local machines, and so on.

  * •

The auxiliary data viewer allows the user to view auxiliary data, such as the
spacecraft locations, velocity, and attitude, using data derived from the
SPICE kernel and pointing solutions computed from aspect system data. The
viewer also provides tools to calculate the looking angles of flares and the
coordinates of solar limbs within the STIX field of view.

  * •

The HK data browser enables users to view time series of all STIX HK
parameters, including the temperature, voltage, operation mode, memory status,
etc. It provides great convenience for the instrument operations team to
monitor the instrument status.

  * •

The STIX data access page offers users a variety of tools to search and
download STIX data products. It also provides links to web tools to preview
the products. As soon as they are generated at the STIX Data Center, STIX data
products are immediately available for access on the page.

###  6.2 STIX Data Center interface via stixdcpy

stixdcpy 111stixdcpy source code is hosted on the GitHub repository at
https://github.com/i4Ds/stixdcpy. is a python package that facilitates
accessing and analyzing STIX data. With stixdcpy, users can easily query and
download the data products available at the STIX Data Center. Similarly to the
web tools, stixdcpy also provides tools to perform some standard analysis of
STIX data, such as dead time correction, transmission correction, data
clipping, and merging. stixdcpy is still under active development. As a
result, its features and capabilities may change over time.

##  7 Summary

STIX is one of ten instruments on board Solar Orbiter, which was launched into
space on 10 February 2020. STIX measures the intensity and spectrum of hard
X-rays emitted during solar flares in the energy range of 4 – 150 keV. During
nominal operations, STIX continuously generates telemetry data. To process and
archive the data as well as to support the operations of the instrument and
scientific activities using STIX data, dedicated data-processing pipelines and
a data platform have been developed at the STIX Data Center. The pipelines
generate telemetry at different levels and perform standard scientific
analyses. The data center platform distributes STIX data products and also
provides users with various web-based tools for searching and browsing STIX
data products. The data center is designed to work in a fully automatic mode
with minimal human intervention. The concept has proven successful and has
been running continuously for more than two years. The data center not only
facilitates the operations of the instrument, but it also provides great
support to STIX data users.

## References

  * Acton et al. (2018) Acton, C., Bachman, N., Semenov, B., & Wright, E. 2018, Planetary and Space Science, 150, 9, enabling Open and Interoperable Access to Planetary Science and Heliophysics Databases and Tools
  * Acton (1996) Acton, C. H. 1996, Planetary and Space Science, 44, 65, planetary data system
  * and (2019) and. 2019, Solar Orbiter SPICE Kernel Dataset
  * Cornwell & Evans (1985) Cornwell, T. J. & Evans, K. F. 1985, A&A, 143, 77
  * Forveille & Shore (2020) Forveille, T. & Shore, S. 2020, A&A, 642, E1
  * Högbom (1974) Högbom, J. A. 1974, A&AS, 15, 417
  * Krucker et al. (2020) Krucker, Hurford, G. J., Grimm, O., et al. 2020, A&A, 642, A15
  * Maier & Limousin (2016) Maier, D. & Limousin, O. 2016, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 812, 43
  * Maier et al. (2020) Maier, D., Limousin, O., & Daniel, G. 2020, EPJ Web Conf., 225, 01003
  * Massa et al. (2022) Massa, P., Battaglia, A. F., Volpara, A., et al. 2022, Sol. Phys., 297, 93
  * Massa et al. (2020) Massa, P., Schwartz, R., Tolbert, A. K., et al. 2020, ApJ, 894, 46
  * Müller et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1
  * Peccia (2005a) Peccia, N. 2005a, in 2005 IEEE Aerospace Conference, 3988–3995
  * Peccia (2005b) Peccia, N. 2005b, in 2005 IEEE Aerospace Conference, 3988–3995
  * Pence, W. D. et al. (2010) Pence, W. D., Chiappetti, L., Page, C. G., Shaw, R. A., & Stobie, E. 2010, A&A, 524, A42
  * Ryan et al. (1988) Ryan, C., Clayton, E., Griffin, W., Sie, S., & Cousens, D. 1988, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 34, 396
  * Skwarnicki (1986) Skwarnicki, T. 1986, PhD thesis, Cracow, INP
  * Volpara et al. (2022) Volpara, A., Massa, P., Perracchione, E., et al. 2022, A&A, 668, A145
  * Warmuth et al. (2020) Warmuth, A., Önel, H., Mann, G., et al. 2020, Sol. Phys., 295, 90


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# Bottomonium Dissociation in a Rotating Plasma

Nelson R. F. Braga  Instituto de Física, Universidade Federal do Rio de
Janeiro, RJ 21941-909, Brazil.  Yan F. Ferreira  Instituto de Física,
Universidade Federal do Rio de Janeiro, RJ 21941-909, Brazil.

(August 28, 2024)

###### Abstract

Heavy vector mesons provide important information about the quark gluon plasma
(QGP) formed in heavy ion collisions. This happens because the fraction of
quarkonium states that are produced depends on the properties of the medium.
The intensity of the dissociation process in a plasma is affected by the
temperature, the chemical potential and the presence of magnetic fields. These
effects have been studied by many authors in the recent years. Another
important factor that can affect the dissociation of heavy mesons, and still
lacks of a better understanding, is the rotation of the plasma. Non central
collisions form a plasma with angular momentum. Here we use a holographic
model to investigate the thermal spectrum of bottomonium quasi-states in a
rotating medium in order to describe how a non vanishing angular velocity
affects the dissociation process.

##  I Introduction

Heavy ion collisions, produced in particle accelerators, lead to the formation
of a new state of matter, a plasma where quarks and gluons are deconfined.
This so called QGP behaves like a perfect fluid and lives for a very short
time[1, 2, 3, 4]. The study of this peculiar state of matter is based on the
analysis of the particles that are observed after the hadronization process
occur and the plasma disappears. For this to be possible, it is necessary to
understand how the properties of the QGP, like temperature ($T$) and density
($\mu$), affect the spectra that reach the detectors. In particular,
quarkonium states, like bottomonium, are very interesting since they survive
the deconfinement process that occur when the QGP is formed. They undergo a
partial dissociation, with an intensity that depends on the characteristics of
the medium, like $T$ and $\mu$. So, it is important to find it out how the
properties of the plasma affect the dissociation.

Bottomonium quasi-states in a thermal medium can be described using
holographic models [5, 6, 7, 8, 9, 10, 11] see also [12, 13, 14]. In
particular, the improved holographic model proposed in [8], which will be
considered here, involves three energy parameters: one representing the heavy
quark mass, another associated with the intensity of the strong interaction
(string tension) and another with the non-hadronic decay of quarkonium. This
model provides good estimates for masses and decay constants.

Besides temperature, density and magnetic fields, another, less studied,
property that affects the thermal behaviour is the rotation of the QGP, that
occurs in non-central collisions. For previous works about the rotation
effects in the QGP, see for example [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32]. In particular, a holographic description of
the QGP in rotation can be found in Refs. [22, 28]. A rotating plasma with
uniform rotational speed is described holographically in these works by a
rotating black hole with cylindrical symmetry. Rotation is obtained by a
coordinate transformation and the holographic model obtained predicts that
plasma rotation decreases the critical temperature of
confinement/deconfinement transition [28]. For a very recent study of
charmonium in a rotating plasma see [33].

The purpose of this work is to study how rotation of the plasma affects the
thermal spectrum of bottomonium quasi-states. In other words, we want to
understand what is the effect of rotation in the dissociation process of
$b\bar{b}$. We will follow two complementary approaches. One is to calculate
the thermal spectral functions and the other is to find the quasinormal models
associated with bottomonium in rotation.

The organization is the following. In section II we present a holographic
model for bottomonium in a rotating plasma. In III we work out the equations
of motion for the fields that describe the quasi states. In IV we discuss the
solutions, taking into account the incoming wave boundary conditions on the
black hole horizon. In section V we calculate the spectral functions for
bottomonium and in VI we present the complex frequencies of the quasi-normal
modes. Finally, section VII contains our conclusions and discussions about the
results obtained.

##  II Holographic Model For Quarkonium in the Plasma

Vector mesons are represented holographically by a vector field
$V_{m}=(V_{t},V_{1},V_{2},V_{3},V_{z})$ , that lives, in the case of a non
rotating plasma, in a five dimensional anti-de Sitter ($\mathrm{AdS}_{5}$)
black hole space with metric

$\displaystyle ḏs²=\dfrac{\raisebox{-1.0pt}{$\displaystyle
R²$}}{\displaystyle{z²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\\!-f(z)ḏt²+(ḏx^{1})²+(ḏx^{2})²+(ḏx^{3})²+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{f(z)}^{\vphantom{\rule{0.0pt}{5.0pt}}}}ḏz²{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)},$
(1) with $\displaystyle f(z)=1-\dfrac{\raisebox{-1.0pt}{$\displaystyle
z^{4}$}}{\displaystyle{z_{h}^{4}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\,.$ (2)

The constant $R$ is the $\mathrm{AdS}$ radius and the Hawking temperature of
the black hole, given by

$\displaystyle T=\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z_{h}}^{\vphantom{\rule{0.0pt}{5.0pt}}}},$ (3)

is identified with the temperature of the plasma.

The action integral for the field has the form

$\displaystyle I=∫ḏ^{4}x∫_{0}^{z_{h}}ḏz\,√{-g}\,\mathcal{L},$ (4) with the
Lagrangian density
$\displaystyle\mathcal{L}=-\dfrac{1}{4g_{5}^{2}}\={e}^{-(z)}g^{mp}g^{nq}F_{mn}F_{pq},$
(5)

where $F_{mn}=\nabla_{\\!m}V_{n}-\nabla_{\\!n}V_{m}=∂_{m}V_{n}-∂_{n}V_{m}$,
with $\nabla_{\\!m}$ being the covariant derivative. The dilaton like
background field $(z)$ is [7, 8]

$\displaystyle(z)=^{2}z^{2}+Mz+\tanh\\!\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\dfrac{1}{Mz}-\dfrac{}{√{}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}.$
(6)

The back reaction of $(z)$ on the metric is not taken into account. This
modified dilaton is introduced in order to obtain an approximation for the
values of masses and decay constants of heavy vector mesons. The three
parameters of the model are fixed at zero temperature to give the best values
for these quantities, specially the decay constants, compared with the
experimental values for bottomonium found in the particle data group table
[34]: ${}_{b}=$2.45\text{\,}\mathrm{GeV}$$,
$√{\smash[b]{{}_{b}}}=$1.55\text{\,}\mathrm{GeV}$$ and
$M_{b}=$6.2\text{\,}\mathrm{GeV}$$. They give the results presented in table
1.

Bottomonium Masses and Decay Constants |  |  |  |
---|---|---|---|---
State | Experimental Masses ($\mathrm{MeV}$) |  $Massesonthetangentmodel($$\mathrm{MeV}$$)$ |  Experimental Decay Constants ($\mathrm{MeV}$) | Decay Constants on the tangent model ($\mathrm{MeV}$)
$1S$ | $ˍ9460.30±0.26$ | ˍ6905 |  $715.0±ˍ4.8$ | 719
$2S$ | $10023.26±0.31$ | ˍ8871 |  $497.4±ˍ4.5$ | 521
$3S$ | $10355.2ˍ±0.5ˍ$ | 10442 |  $430.1±ˍ3.9$ | 427
$4S$ | $10579.4ˍ±1.2ˍ$ | 11772 |  $341ˍ\;±18ˍ\;$ | 375

Table 1: Comparison of bottomonium masses and decay constants obtained
experimentally [34] and from the tangent model.

In order to have some characterization of the quality of the fit, one can
define the root mean square percentage error (RMSPE) as

$\displaystyle\textrm{RMSPE}=100\%×√{\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{N-N_{p}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\sum_{i\,=\,1}^{N}\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\dfrac{\raisebox{-1.0pt}{$\displaystyle
y_{i}-\hat{y}_{i}$}}{\displaystyle{\hat{y}_{i}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!2}},$
(7)

where $N=8$ is the number of experimental points (4 masses and 4 decay
constants), $N_{p}=3$ is the number of parameters of the model, the $y_{i}$’s
are the values of masses and decay constants predicted by the model and the
$\hat{y}_{i}$’s are the experimental values of masses and decay constants.
With this definition, we have $\textrm{RMSPE}=14.8\%$ for bottomonium.

Now, in order to analyse the case of a rotating plasma, with homogeneous
angular velocity, we consider an $AdS_{5}$ space with cylindrical symmetry by
writing the metric as

$\displaystyle ḏs²=\dfrac{\raisebox{-1.0pt}{$\displaystyle
R²$}}{\displaystyle{z²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\\!-f(z)ḏt²+ℓ²ḏ²+(ḏx^{1})²+(ḏx^{2})²+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{f(z)}^{\vphantom{\rule{0.0pt}{5.0pt}}}}ḏz²{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)},$
(8)

where $R$ is again the AdS radius and $ℓ$ is the hypercylinder radius.

As in ref. [23, 22, 28], we introduce rotation via the Lorentz-like coordinate
transformation

$\displaystyle\begin{aligned} t&\-->(t+ℓ²)\\\ &\-->(t+),\end{aligned}$ (9)
with $\displaystyle=\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{√{1-²ℓ²}}^{\vphantom{\rule{0.0pt}{5.0pt}}}},$ (10)

where is the angular velocity of the rotation. With this transformation, the
metric (8) becomes

$\displaystyle\begin{split}ḏs^{2}=\dfrac{\raisebox{-1.0pt}{$\displaystyle
R^{2}$}}{\displaystyle{z^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.-²\MULTIPLY{0.75}{2}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}f(z)-²ℓ²{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}ḏt^{2}+2²\MULTIPLY{0.75}{2}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-f(z){\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}ℓ²ḏtḏ+²\MULTIPLY{0.75}{2}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-²ℓ²f(z){\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}ℓ²ḏ^{2}\qquad\\\
+\;(ḏx^{1})^{2}+(ḏx^{2})^{2}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{f(z)}^{\vphantom{\rule{0.0pt}{5.0pt}}}}ḏz^{2}\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right],\end{split}$
(11)

The temperature of the rotating AdS black hole is[23, 22, 28]

$\displaystyle T=\dfrac{1}{z_{h}}√{1-²ℓ²}.$ (12)

Note that we recover (8) by doing $-->0$ in (11).

We assume that the rotating black hole metric of Eq. (11) is dual to a
cylindrical slice of rotating plasma is dual to a cylindrical slice of
rotating plasma. This interpretation can be justified by analysing the angular
momentum $J$. For the metric (11), $J$ can be calculated [28] with the result

$J=-\frac{\partial\Phi}{\partial\Omega}=\frac{2L^{3}}{\kappa^{2}}\frac{\Omega}{z_{h}^{4}(1-\Omega^{2}l^{2})}\;,$
(13)

while for the metric (8) one has $J=0$. Thus, the coordinate transformation is
adding angular momentum to the system and therefore representing a plasma in
rotation.

##  III Equations of Motion

By extremizing the action (4), we find the equations of motion

$\displaystyle∂_{n}(√{-g}\={e}^{-}F^{mn})=0.$ (14)

We now choose a Fourier component of the field and, for simplicity, consider
zero momentum (meson at rest), $V_{m}(t,\bm{x},z)=v_{m}(,z)\={e}^{-\^{\i}t}$.
We also choose the gauge $V_{z}=0$. The equations of motion (14) become

$\displaystyle\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-²ℓ²f$}}{\displaystyle{1-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle²$}}{\displaystyle{f²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{i}+\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}v_{i}^{\prime}+v_{i}^{\prime\prime}=0\qquad(i=1,2),$
(15)
$\displaystyle\begin{multlined}-\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{f-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{-1}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}ℓ²v_{t}^{\prime}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{f-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}ℓ²v_{t}^{\prime\prime}\\\
+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-²ℓ²$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle²$}}{\displaystyle{f²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{+}\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{^{\prime}}+v_{^{\prime\prime}}=0,\end{multlined}-\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{f-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{-1}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}ℓ²v_{t}^{\prime}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{f-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}ℓ²v_{t}^{\prime\prime}\\\
+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-²ℓ²$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle²$}}{\displaystyle{f²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{+}\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{1-²ℓ²f^{-1}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{^{\prime}}+v_{^{\prime\prime}}=0,$
(18)
$\displaystyle\begin{multlined}\phantom{-}\,\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{f^{-1}-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{-1}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{^{\prime}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{^{\prime\prime}}\\\
-\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{(²ℓ²f)^{-1}-1}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{t}^{\prime}+v_{t}^{\prime\prime}=0\;\end{multlined}\phantom{-}\,\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{f^{-1}-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{-1}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{^{\prime}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{^{\prime\prime}}\\\
-\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{(²ℓ²f)^{-1}-1}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+^{\prime}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{t}^{\prime}+v_{t}^{\prime\prime}=0\;$
(21) and $\displaystyle v_{t}^{\prime}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{^{\prime}}=0.$
(22)

where the prime stands for the derivative with respect to $z$, $f^{-1}=1/f$,
and, for simplicity, we omit the dependence on $z$ of $f$ and and the
dependence on $(,z)$ of the fields $v_{\textmu}$. These equations are not all
independent, if we substitute (22) into (21) we obtain an identity, and
substituting the same equation into (18), we obtain an equation for $v$ only.
With this, the system of equations of motion simplifies to

$\displaystyle\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-²ℓ²f$}}{\displaystyle{1-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle²$}}{\displaystyle{f²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{i}+\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}v_{i}^{\prime}+v_{i}^{\prime\prime}=0\qquad(i=1,2),$
(23) $\displaystyle\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-²ℓ²f$}}{\displaystyle{1-²ℓ²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle²$}}{\displaystyle{f²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{+}\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f^{\prime}$}}{\displaystyle{f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}v_{^{\prime}}+v_{^{\prime\prime}}=0,$
(24) $\displaystyle v_{t}^{\prime}-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1-f$}}{\displaystyle{1-²ℓ²f}^{\vphantom{\rule{0.0pt}{5.0pt}}}}v_{^{\prime}}=0.$
(25)

Note that when $ℓ=0$, equations (23) and (24) are the same and the two states
are degenerate. Rotation breaks this degeneracy.

From this point on, we will divide our analysis in two cases, according to
three possible polarizations. The first case is for polarizations in
directions $x^{1}$ and $x^{2}$, for which $(v_{m})=(0,v,0,0,0)$ or
$(v_{m})=(0,0,v,0,0)$. The second case is for the polarization in direction ,
for which $(v_{m})=(0,0,0,v,0)$.

##  IV Solving the Equations of Motion

###  IV.1 Near the horizon behavior of the solution

By approximating the function $f$ as the first term of its power series at
$z=z_{h}$, we write

$\displaystyle f(z)=^{\prime}f^{\prime}(z_{h})(z-z_{h})\qquad\text{(for
$z=^{\prime}z_{h}$)}$ (26)

and we see that, in this region, equations (23) and (24) both take the form

$\displaystyle\dfrac{{}^{2}{}^{2}}{\displaystyle\rule{0.0pt}{11.5pt}f^{\prime}(z_{h})^{2}(z-z_{h})^{2}}\,v_{\mathrm{hor}}(z)+\dfrac{1}{z-z_{h}}\,v_{\mathrm{hor}}^{\prime}(z)+\,v_{\mathrm{hor}}^{\prime\prime}(z)=0,$
(27)

which, in terms of the temperature, can be written as

$\displaystyle\dfrac{{}^{2}}{\displaystyle\rule{0.0pt}{11.5pt}(4T)^{2}}\,v_{\mathrm{hor}}(z)+(z-z_{h})\,v_{\mathrm{hor}}^{\prime}(z)+(z-z_{h})^{2}\,v_{\mathrm{hor}}^{\prime\prime}(z)=0.$
(28)

The two solutions of this equation are

$\displaystyle\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{z}{z_{h}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!-\^{\i}/4T}\qquad\qquad\text{and}\qquad\qquad\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{z}{z_{h}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!+\^{\i}/4T}.$
(29)

The solution with the minus sign in the exponent corresponds to an infalling
wave at the horizon, while the other, with positive sign, corresponds to an
outgoing wave. This becomes clear if one change to the Regge-Wheeler tortoise
coordinate, as explained in refs. [35, 36].

The black hole allows only infalling waves at the horizon. Therefore, the
field that solves the complete equations of motion has to obey the condition

$\displaystyle
v(z)=^{\prime}A\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{\raisebox{-1.0pt}{$\displaystyle
z$}}{\displaystyle{z_{h}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!-\^{\i}/4T}\qquad\text{(for
$z=^{\prime}z_{h}$)},$ (30)

where $A$ is a normalization constant. The norm of the field will have no
importance for us, then we can set $A=1$.

In order to solve the complete equations of motion (23) and (24) numerically,
we translate the infalling wave condition at the horizon in two boundary
conditions to be imposed at a point $z=z_{0}$ close to the horizon:

$\displaystyle v(z_{0})=v_{\mathrm{hor},\,p}(z_{0})\qquad\text{and}\qquad
v^{\prime}(z_{0})=v_{\mathrm{hor},\,p}^{\prime}(z_{0}),$ (31) with
$\displaystyle
v_{\mathrm{hor},\,p}(z)=\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{z}{z_{h}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!-\^{\i}/4T}\sum_{n=0}^{p}a_{n}\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{z}{z_{h}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}^{\\!\\!n}.$
(32)

The function $v_{\mathrm{hor},\,p}(z)$ is just the infalling wave expression
(30) (with $A$ set to 1) times a polynomial correction of order $p$ introduced
for purposes of the numerical calculations. The coefficient $a_{0}$ is, of
course, 1 and the other coefficients $a_{n}$ are determined by imposing the
equation of motion (23) or (24) with $v=v_{\mathrm{hor},\,p}$ to be valid up
to the order $p$. The coefficients $a_{n}$ are, therefore, the same for
polarization in directions $x^{1}$ and $x^{2}$ but are different from the ones
for polarization in the direction . The point $z_{0}$ is a choosen value of
$z$, close to $z_{h}$, where the approximation
$v(z)=^{\prime}v_{\mathrm{hor},\,p}(z)$ is valid. Using equation (30), the
leading term of the field, for $z$ close to $z_{h}$, can be written as:

$\displaystyle\cos\\!\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle$}}{\displaystyle{4T}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\ln\\!\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{\raisebox{-1.0pt}{$\displaystyle
z$}}{\displaystyle{z_{h}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}-\^{\i}\sin\\!\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle$}}{\displaystyle{4T}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\ln\\!\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}1-\dfrac{\raisebox{-1.0pt}{$\displaystyle
z$}}{\displaystyle{z_{h}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}.$
(33)

In the region of values of $z$ close to the horizon, very small changes in $z$
produce significant changes in $v(z)$, a problem for the numerical
calculation. For this reason, the point $z_{0}$ cannot be chosen too close to
$z_{h}$. This is the reason why we introduce the polynomial perturbation from
(32) in the infalling condition. In this work, the value $z_{0}=0.9z_{h}$ with
20 coefficients $a_{n}$ was sufficient.

Figure 1: Real (left) and Imaginary (right) parts of the field $v(,z)$ (blue)
and its approximations near the horizon $v_{\mathrm{hor},\,p}(,z)$ with 20
(orange) and 10 (green) coefficients for the representative value of
$=$10\text{\,}\mathrm{GeV}$$ for the non-rotating plasma at temperature
$T=$150\text{\,}\mathrm{MeV}$$. In the first line we plot the field in all its
domain. In the second line we zoom in the region from $z=0.7z_{h}$ to
$z=z_{h}$. The vertical lines highlight the values $z=0.9z_{h}$ and
$z=0.99z_{h}$.

Figure 1 shows the solution of the equation of motion for the non-rotating
plasma at a specific temperature and for some representative value of  as well
as two approximations for this solution near the horizon. From this figure,
one can see that if we had chosen $z_{0}=0.99z_{h}$, for example, instead of
$z_{0}=0.9z_{h}$, we would be in the unstable region and any small numerical
error would be significantly propagated. Also, if we had used 10 coefficients,
for example, instead of 20, we woldn't have a good aproximation for the field
at $z_{0}=0.9z_{h}$. For more discussion on this method, see, for example,
[37].

This method produces a numerical solution of equations (23) or (24) with the
infalling wave condition at $z_{h}$ for any value of . We will use these
solutions to calculate spectral functions and quasinormal modes in the
following sections.

##  V Spectral Functions

Spectral Functions are defined in terms of the retarded Green's functions as

${}_{\textmu}()=-2\,\textrm{Im}\,G_{\textmu}^{R}().$ (34)

They provide an important way of analysing the dissociation of quarkonia in a
thermal medium. At zero temperature, the spectral function of a quarkonium,
considering just one particle states, is a set of delta peaks at the values of
the holographic masses of table 1. At finite temperature, these peaks acquire
a finite height and a non-zero width. As the temperature increases the height
of each peak decreases and its width increases. This broadening effect of the
peaks indicates dissociation in the medium. In this section we calculate the
spectral function for bottomonium in a rotating plasma at three fixed
temperatures in order to analyse the effect of the rotational speed in the
dissociation process.

###  V.1 Retarded Green's Function

In the four dimensional vector gauge theory we define a retarded Green’s
functions of the currents $J$, that represent the heavy vector mesons, as

$\displaystyle
G_{\textmu}^{R}=-\^{\i}∫ḏ^{4}x\={e}^{-\^{\i}p\textperiodcentered
x}(t)\MULTIPLY{0.75}{0}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left<\rule{0.0pt}{0.0pt}}\right.}[J_{\textmu}(x),J_{(}0)]{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right>}.$
(35)

The Son-Starinets prescription [38] provide a way of extracting the retarded
Green's function from the on shell action of the dual vector fields in AdS
space

$\displaystyle I_{\text{on shell}}$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{4g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ^{4}x∫_{0}^{z_{h}}ḏz\,√{-g}\,\={e}^{-(z)}F_{mn}F^{mn}$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ^{4}x∫_{0}^{z_{h}}ḏz\,∂_{m}(√{-g}\,\={e}^{-(z)}V_{n}F^{mn}),$
(36)

where we have used the equations of motion, (14), to go from the first line to
the second.

In the gauge $V_{z}=0$, we have $V_{n}F^{mn}=V_{F}^{m}$ and, therefore,

$\displaystyle I_{\text{on shell}}$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\begin{multlined}\MULTIPLY{0.75}{3.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏx¹ḏx²ḏḏz√{-g}\,\={e}^{-(z)}V_{F}^{t}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{t\,=\,-∞}^{t\,=\,+∞}+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx²ḏḏz\,√{-g}\,\={e}^{-(z)}V_{F}^{1}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{x^{1}\,=\,-∞}^{x^{1}\,=\,+∞}\\\\[0.0pt]
+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx¹ḏḏz\,√{-g}\,\={e}^{-(z)}V_{F}^{2}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{x^{2}\,=\,-∞}^{x^{2}\,=\,+∞}\;+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx¹ḏx²ḏz\,√{-g}\,\={e}^{-(z)}V_{F}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{\,=\,0}^{\,=\,2}\\\\[0.0pt]
+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏ^{4}x\,√{-g}\,\={e}^{-(z)}V_{F}^{z}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}\,\MULTIPLY{0.75}{3.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right].\end{multlined}\MULTIPLY{0.75}{3.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏx¹ḏx²ḏḏz√{-g}\,\={e}^{-(z)}V_{F}^{t}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{t\,=\,-∞}^{t\,=\,+∞}+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx²ḏḏz\,√{-g}\,\={e}^{-(z)}V_{F}^{1}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{x^{1}\,=\,-∞}^{x^{1}\,=\,+∞}\\\\[0.0pt]
+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx¹ḏḏz\,√{-g}\,\={e}^{-(z)}V_{F}^{2}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{x^{2}\,=\,-∞}^{x^{2}\,=\,+∞}\;+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏtḏx¹ḏx²ḏz\,√{-g}\,\={e}^{-(z)}V_{F}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{\,=\,0}^{\,=\,2}\\\\[0.0pt]
+\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏ^{4}x\,√{-g}\,\={e}^{-(z)}V_{F}^{z}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}\,\MULTIPLY{0.75}{3.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right].$
(40)

Since any physical field has to go to zero as $t$, $x^{1}$ or $x^{2}$ goes to
$±∞$, the first three terms vanish. The point with $=0$ is equivalent to the
point $=2$, therefore, the fourth term vanishes too. This leaves us just with
the surface term

$\displaystyle I_{\text{on shell}}$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,∫ḏ^{4}x\,√{-g}\,\={e}^{-(z)}V_{F}^{z}\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.$
(41)

In momentum space and considering the meson at rest, we find

$\displaystyle I_{\text{on shell}}=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ\,\MULTIPLY{0.75}{2.5}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,√{-g}\,\={e}^{-(z)}g^{zz}g^{\textmu}v_{\textmu}(-,z)v_{^{\prime}}(,z)\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.$
(42)

Using the equation of motion (22) to substitute $v_{t}^{\prime}$ in terms of
$v_{^{\prime}}$, one eliminates $v_{t}$ ending up with

$\displaystyle\begin{multlined}I_{\text{on
shell}}=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ\,√{-g}\,\={e}^{-(z)}g^{zz}\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left\\{\rule{0.0pt}{0.0pt}}\right.\sum_{j=1,2}g^{jj}v_{j}(-,z)v_{j}^{\prime}(,z)\\\\[-3.60004pt]
+\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle(g^{t})²$}}{\displaystyle{-g^{tt}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+g\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{(}-,z)v_{^{\prime}}(,z)\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right\\}\MULTIPLY{0.75}{4}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.\end{multlined}I_{\text{on
shell}}=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ\,√{-g}\,\={e}^{-(z)}g^{zz}\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left\\{\rule{0.0pt}{0.0pt}}\right.\sum_{j=1,2}g^{jj}v_{j}(-,z)v_{j}^{\prime}(,z)\\\\[-3.60004pt]
+\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle(g^{t})²$}}{\displaystyle{-g^{tt}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+g\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}v_{(}-,z)v_{^{\prime}}(,z)\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right\\}\MULTIPLY{0.75}{4}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.$
(45)

Now we separate the value of the field at the boundary $z=0$ by defining the
bulk to boundary propagator $\mathcal{E}_{\textmu}(,z)$ such that

$\displaystyle
v_{\textmu}(,z)=\mathcal{E}_{\textmu}(,z)v^{0}_{\textmu}()\qquad\qquad\text{(no
summation, $\textmu=1,2,$),}$ (46)

with $v^{0}_{\textmu}()=v_{\textmu}(,0)$. This implies the bulk to boundary
condition $\mathcal{E}_{\textmu}(,0)=1$. Using the definition (46) in equation
(45), the on shell action becomes

$\displaystyle\begin{multlined}I_{\text{on
shell}}=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ\,√{-g}\,\={e}^{-(z)}g^{zz}\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left\\{\rule{0.0pt}{0.0pt}}\right.\sum_{j=1,2}g^{jj}\mathcal{E}_{j}(-,z)v_{j}^{0}(-)\mathcal{E}_{j}^{\prime}(,z)v_{j}^{0}()\\\\[-1.20007pt]
+\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle(g^{t})²$}}{\displaystyle{-g^{tt}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+g\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}\mathcal{E}_{(}-,z)v_{^{0}}(-)\mathcal{E}_{^{\prime}}(,z)v_{^{0}}()\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right\\}\MULTIPLY{0.75}{4}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.\end{multlined}I_{\text{on
shell}}=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{2g_{5}^{2}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}∫ḏ\,√{-g}\,\={e}^{-(z)}g^{zz}\,\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left\\{\rule{0.0pt}{0.0pt}}\right.\sum_{j=1,2}g^{jj}\mathcal{E}_{j}(-,z)v_{j}^{0}(-)\mathcal{E}_{j}^{\prime}(,z)v_{j}^{0}()\\\\[-1.20007pt]
+\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\mathopen{}\mathclose{{}\left[\rule{0.0pt}{0.0pt}}\right.}\,\dfrac{\raisebox{-1.0pt}{$\displaystyle(g^{t})²$}}{\displaystyle{-g^{tt}}^{\vphantom{\rule{0.0pt}{5.0pt}}}}+g\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right]}\mathcal{E}_{(}-,z)v_{^{0}}(-)\mathcal{E}_{^{\prime}}(,z)v_{^{0}}()\MULTIPLY{0.75}{3}{\tempa}\ADD{\tempa}{0.50}{\tempb}\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right\\}\MULTIPLY{0.75}{4}{\tempa}\ADD{\tempa}{0.50}{\tempb}{\\!\\!\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right.}\,\,{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right|}_{z\,=\,0}^{z\,=\,z_{h}}.$
(49)

Then, applying the Son-Starinets prescription, we determine the retarded
Green's functions

$\displaystyle G_{jj}^{R}()$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
ℓR$}}{\displaystyle{g_{5}²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\={e}^{-(0)}\lim_{z-->0}\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\mathcal{E}_{j}^{\prime}(,z)\qquad\text{(no
summation, $j=1,2$)}$ (50) and $\displaystyle G^{R}()$
$\displaystyle=-\dfrac{\raisebox{-1.0pt}{$\displaystyle
R$}}{\displaystyle{ℓg_{5}²}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\={e}^{-(0)}\lim_{z-->0}\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}\mathcal{E}_{^{\prime}}(,z).$
(51)

The other $G_{\textmu}^{R}$ vanish.

In vacuum ($T=0$), the Green's function is just

$\displaystyle(p²)=\sum_{n=1}^{∞}\dfrac{\raisebox{-1.0pt}{$\displaystyle
f_{n}^{2}$}}{\displaystyle{-p²-
m_{n}²+\^{\i}}^{\vphantom{\rule{0.0pt}{5.0pt}}}},$ (52)

where $m_{n}$ and $f_{n}$ are the mass and decay constant of the radial states
of excitation level $n$ of bottomonium. The imaginary part of this Green's
function and, hence, the spectral function at zero temperature is proportional
to

$\displaystyle\sum_{n=1}^{∞}f_{n}(p²+m_{n}²),$ (53)

a set of delta peaks, each one located at the mass $m_{n}$ of a state.

When the meson is inside a thermal medium, at a non-zero temperature, the
change in the spectral function is a broadening of the peaks. These peaks
acquire a finite height and a non zero width. This broadening effect rises
with the temperature and with the excitation level $n$ and is interpreted as
dissociation in the thermal medium. Variations of the spectral function of
quarkonia in a thermal medium without rotation can be found on [8, 9, 10, 11].
The same holographic model considered here was used in these references.

###  V.2 Numerical Results of Spectral Functions

Figure 2: Spectral functions ${}_{11}()=_{22}()$ and $()$ for different values
of rotation speed $ℓ$ and temperature fixed at $T=$150\text{\,}\mathrm{MeV}$$.
Figure 3: Spectral functions ${}_{11}()=_{22}()$ and $()$ for different values
of rotation speed $ℓ$ and temperature fixed at $T=$200\text{\,}\mathrm{MeV}$$.
Figure 4: Spectral functions ${}_{11}()=_{22}()$ and $()$ for different values
of rotation speed $ℓ$ and temperature fixed at $T=$250\text{\,}\mathrm{MeV}$$.

Figure 2 shows how bottomonium's spectral function changes with the rotation
speed $ℓ$ at temperature $T=$150\text{\,}\mathrm{MeV}$$. Figures 3 and 4 do
the same for temperatures fixed at $T=$200\text{\,}\mathrm{MeV}$$ and
$T=$250\text{\,}\mathrm{MeV}$$, respectively. In these charts we multiplied
the spectral functions by the inverse of the constants that appear in eqs.
(50) and (51) in order to represent functions with the same dimension, that
can be compared. From these figures one can see that rotation increases the
dissociation effect and also that fields with polarization $v_{1}$ and $v_{2}$
dissociate slightly faster than the ones with polarization $v$.

##  VI Quasinormal Modes

In vacuum, the equations of motion simplify to

$\displaystyle²v+\MULTIPLY{0.75}{3}{\vara}\ADD{\vara}{0.50}{\varb}{\mathopen{}\mathclose{{}\left(\rule{0.0pt}{0.0pt}}\right.}\\!\\!-\dfrac{\raisebox{-1.0pt}{$\displaystyle
1$}}{\displaystyle{z}^{\vphantom{\rule{0.0pt}{5.0pt}}}}-^{\prime}{\mathopen{}\mathclose{{}\left.\rule{0.0pt}{0.0pt}}\right)}v^{\prime}+v^{\prime\prime}=0.$
(54)

In this case, there is no black hole and, therefore, no infalling wave
condition. We determine the normal modes by solving these equations with the
exigence of the field to satisfy the normalization condition

$\displaystyle∫_{0}^{∞}\dfrac{R}{z}\={e}^{-(z)}\,|v(z)|²\,ḏz=1.$ (55)

It is possible to translate this normalization condition into the Dirichlet
condition

$\displaystyle v(,z=0)=0.$ (56)

This equation is solvable only for a discrete set of real values n. These
values are the masses of the quarkonium states in vacuum. They are shown in
the third column of table 1.

At finite temperature, instead of normal modes, we have the quasinormal modes.
They are the solutions of the equations of motion (23) and (24), that satisfy

  1. 1.

the infalling wave condition at the horizon,

  2. 2.

the Dirichlet condition (56).

The values n that satisfy both of this conditions are called quasinormal
frequencies, the fields $v_{\textmu}(_{n},z)$ are called quasinormal modes and
represent the meson quasistates.

As the value at $T=0$ of the normal frequency n is interpreted as the mass of
the particle in its state $n$, the real part of the quasinormal frequency
$\textrm{Re}(_{n})$, at finite temperature, is interpreted as the thermal mass
of the quasiparticle. The imaginary part $\textrm{Im}(_{n})$ is related to its
degree of dissociation. The larger the absolute value of the imaginary part,
the stronger the dissociation. It is interesting to note that the real and
imaginary parts of a $n$-th quasinormal mode are related to the position and
width of the $n$-th peak in the spectral function. Therefore, we can interpret
a growth in the imaginary part of the quasinormal frequency as an increase in
the dissociation effect. Indeed, at $T=0$ the width of the spectral function
peaks is zero, as is the imaginary part of the frequency n. Also, the limit
for $T->0$ of $\textrm{Re}(_{n})$ is the mass $m_{n}$. This discussion for the
non-rotating plasma with temperature only is already present in the
literature. One can find an application of the tangent model for this case on
Refs.[8, 9, 10, 11].

The results of quasinormal frequencies for polarizations $x^{j}$ and  as
function of the rotation speed $ℓ$ and for temperature fixed at
$T=$200\text{\,}\mathrm{MeV}$$ are shown in figure 5. From this figure, one
sees that the dissociation degree, measured by $-\textrm{Im}()$, rises with
the rotation speed $ℓ$.

Figure 5: Quasinormal frequencies of the different excitation levels as
functions of the rotation speed $ℓ$ and temperature fixed at
$T=$200\text{\,}\mathrm{MeV}$$.

##  VII Conclusions

We analysed in this work how does rotation of a quark gluon plasma affect the
dissociation of heavy vector mesons that are inside the medium. The motivation
for such a study is the fact that non central heavy ion collisions lead to the
formation of a QGP with high angular momentum. So, a description of quarkonium
inside the plasma should take rotation into account. We considered, as an
initial study, the case of a cylindrical shell of plasma in rotation about the
symmetry axis. The real case of the QGP should involve a volume rather than a
cylindrical surface and also possible interaction between different layers of
the plasma, that would have different rotational speeds. However this simple
case considered here already provides important non trivial information. It is
clear from the results obtained here that rotation enhances the dissociation
process for heavy vector mesons inside a plasma. It was also found that these
effect, caused by rotation, is more intense for heavy vector mesons that have
polarization perpendicular to the rotation axis.

Acknowledgments: N.R.F.B. is partially supported by CNPq — Conselho Nacional
de Desenvolvimento Científico e Tecnologico grant 307641/2015-5 and by FAPERJ
— Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de
Janeiro. The authors received also support from Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior — Brasil (CAPES), Finance Code 001.

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