11institutetext:  Max-Planck-Institut für extraterrestrische Physik,
Gießenbachstraße 1, 85748 Garching, Germany
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institut für
Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
33institutetext: International Centre for Radio Astronomy Research, Curtin
University, GPO Box U1987, Perth, WA 6845, Australia 44institutetext: Leibniz-
Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany

# Rapid evolution of the recurrence time in the repeating partial tidal
disruption event eRASSt J045650.3$-$203750

Zhu Liu Rapid evolution of the recurrence time in the repeating partial tidal
disruption event eRASSt J045650.3$-$203750Rapid evolution of the recurrence
time in the repeating partial tidal disruption event eRASSt J045650.3$-$203750
Taeho Ryu Rapid evolution of the recurrence time in the repeating partial
tidal disruption event eRASSt J045650.3$-$203750Rapid evolution of the
recurrence time in the repeating partial tidal disruption event eRASSt
J045650.3$-$203750 A. J. Goodwin Rapid evolution of the recurrence time in the
repeating partial tidal disruption event eRASSt J045650.3$-$203750Rapid
evolution of the recurrence time in the repeating partial tidal disruption
event eRASSt J045650.3$-$203750 A. Rau Rapid evolution of the recurrence time
in the repeating partial tidal disruption event eRASSt J045650.3$-$203750Rapid
evolution of the recurrence time in the repeating partial tidal disruption
event eRASSt J045650.3$-$203750 D. Homan Rapid evolution of the recurrence
time in the repeating partial tidal disruption event eRASSt
J045650.3$-$203750Rapid evolution of the recurrence time in the repeating
partial tidal disruption event eRASSt J045650.3$-$203750 M. Krumpe Rapid
evolution of the recurrence time in the repeating partial tidal disruption
event eRASSt J045650.3$-$203750Rapid evolution of the recurrence time in the
repeating partial tidal disruption event eRASSt J045650.3$-$203750 A. Merloni
Rapid evolution of the recurrence time in the repeating partial tidal
disruption event eRASSt J045650.3$-$203750Rapid evolution of the recurrence
time in the repeating partial tidal disruption event eRASSt J045650.3$-$203750
I. Grotova Rapid evolution of the recurrence time in the repeating partial
tidal disruption event eRASSt J045650.3$-$203750Rapid evolution of the
recurrence time in the repeating partial tidal disruption event eRASSt
J045650.3$-$203750 G. E. Anderson Rapid evolution of the recurrence time in
the repeating partial tidal disruption event eRASSt J045650.3$-$203750Rapid
evolution of the recurrence time in the repeating partial tidal disruption
event eRASSt J045650.3$-$203750 A. Malyali Rapid evolution of the recurrence
time in the repeating partial tidal disruption event eRASSt
J045650.3$-$203750Rapid evolution of the recurrence time in the repeating
partial tidal disruption event eRASSt J045650.3$-$203750 J. C. A. Miller-Jones
Rapid evolution of the recurrence time in the repeating partial tidal
disruption event eRASSt J045650.3$-$203750Rapid evolution of the recurrence
time in the repeating partial tidal disruption event eRASSt J045650.3$-$203750

(Received xxx xx, xxxx; accepted xxx xx, xxxx)

In this letter, we present the results from further X-ray and UV observations
of the nuclear transient eRASSt J045650.3$-$203750 (hereafter J0456$-$20). We
detected five repeating X-ray and UV flares from J0456$-$20, making it one of
the most promising repeating partial tidal disruption event (pTDE) candidates.
More importantly, we also found rapid changes in the recurrence time
$T_{\text{recur}}$ of the X-ray flares by modelling the long-term X-ray light
curve of J0456$-$20\. $T_{\text{recur}}$ first decreased rapidly from about
300 days to around 230 days. It continued to decrease to around 190 days with
an indication of a constant $T_{\text{recur}}$ evidenced from the latest three
cycles. Our hydrodynamic simulations suggest that, in the repeating pTDE
scenario, such rapid evolution of $T_{\text{recur}}$ could be reproduced if
the original star is a $1\,\mathrm{M}_{\odot}$ main-sequence star near the
terminal age and loses nearly 80–90% of its mass during the initial encounter
with a supermassive black hole (SMBH) of mass around
$10^{5}\leavevmode\nobreak\ \mathrm{M}_{\odot}$. The inferred mass loss of
0.8–0.9 $\mathrm{M}_{\odot}$ is higher than the estimated value of around 0.12
$\mathrm{M}_{\odot}$ from observation, which could be explained if the
radiation efficiency is low (i.e. $\ll 0.1$). Our results indicate that
repeating pTDEs could be effective tools to explore the dynamics around
supermassive black holes beyond our own Galaxy.

###### Key Words.:

X-rays: individuals: eRASSt J045650.3$-$203750 – Accretion, accretion disks –
Galaxies: nuclei – Black hole physics

##  1 Introduction

Tidal disruption events (TDEs) are typically considered one-off events where a
star is completely destroyed by a supermassive black hole (SMBH) at the first
pericenter passage. However, theoretical calculations and numeric simulations
have shown that a partial TDE (pTDE) can also occur (Ryu et al. 2020). In a
pTDE, the star loses only a fraction of its mass and survives its first
encounter with the SMBH. If the star is initially in a bound orbit with low
eccentricity, it is expected to generate repeating flares (Hayasaki et al.
2013; Ryu et al. 2020). Because the encounter cross sections of pTDE are
generally larger than or comparable to those of full TDEs, the rate of pTDE is
expected to be larger or comparable to full TDEs (Krolik et al. 2020; Bortolas
et al. 2023). Repeating pTDEs are particularly interesting as they may be
effective probes to explore stellar dynamics around SMBHs beyond our own
Galaxy and as they provide ideal laboratories to study the accretion processes
in SMBHs.

Only a few repeating pTDE candidates have been reported so far (e.g.
ASASSN-14ko, Payne et al. 2021; HLX-1 111Note that HLX-1 is believed to be an
intermediate mass black hole (IMBH) with mass of $\sim
10^{4-5}\mathrm{M}_{\odot}$ (Davis et al. 2011; Webb et al. 2012), Webb et al.
2023; eRASSt J045650.3$-$203750, hereafter J0456$-$20, Liu et al. 2023; RX
J133157.6$-$324319.7, Malyali et al. 2023; AT2018fyk, Wevers et al. 2023).
Among these, J0456$-$20 is one of the best-studied repeating nuclear
transients discovered in a quiescent galaxy ($z=0.077$). Liu et al. (2023)
reported the detection of three repeated X-ray and UV flares from J0456$-$20\.
In particular, the profiles of the X-ray flares are similar and can be
characterised by four distinctive phases: an X-ray rising phase
(${P}_{\text{X, rise}}$) leading into an X-ray plateau phase (${P}_{\text{X,
plat}}$), which is terminated by a rapid X-ray drop phase (${P}_{\text{X,
drop}}$), and followed by an X-ray faint state (${P}_{\text{X, faint}}$).
These results provide strong evidence that J0456$-$20 is a repeating nuclear
transient, making J0456$-$20 one of the most promising pTDE candidates.

Interestingly, a recent study indicates a potential connection between
repeating pTDE and the so-called quasi-periodic eruptions (QPEs) phenomenon
(Miniutti et al. 2023), which exhibits X-ray eruptions on time-scale of less
than one day (Miniutti et al. 2019; Giustini et al. 2020; Arcodia et al. 2021;
Evans et al. 2023; Guolo et al. 2023)). It is thus interesting to study the
evolution of $T_{\text{recur}}$ of the flares for repeating pTDEs. However,
substantial evolution in $T_{\text{recur}}$ has only been reported in
ASASSN-14ko and HLX-1. A period derivative of $-0.0026\pm 0.0006$, with a
period of $115.2^{+1.3}_{-1.2}\leavevmode\nobreak\ $days, has been reported
for ASASSN-14ko (Payne et al. 2022). HLX-1 initially shows quasi-periodic
X-ray outbursts spaced by about $1$ yr between 2009 and 2012. The
$T_{\text{recur}}$ then started to increase until 2018, after which no X-ray
outbursts were detected (Godet et al. 2014; Webb et al. 2023). Long-term
multi-wavelength observations on a larger sample of repeating pTDEs are
required to fully understand the evolution of $T_{\text{recur}}$ in these
objects and potentially provide further evidence to confirm the relation
between repeating pTDEs and QPEs.

In this letter, we report the discovery of the rapid evolution of
$T_{\text{recur}}$ in J0456$-$20\. A detailed analysis of the latest X-ray and
UV data reveals five X-ray and UV flares in J0456$-$20 (marked as cycle 1–5 in
Fig. 1). These results confirm that J0456$-$20 is indeed a repeating nuclear
transient. In addition, they also suggest an initially rapid decrease in
$T_{\text{recur}}$ by more than two months between cycles 1 and 2, slowing to
around $40\leavevmode\nobreak\ $days between cycles 2 and 3.
$T_{\text{recur}}$ reaches an almost constant value of around
$190\leavevmode\nobreak\ $days in the latest three cycles. This paper is
structured as follows. In Sect. 2, we present the multi-wavelength data
reduction. X-ray spectral and light curve modelling are presented in Sect. 3.
Finally, we discuss and summarise our results in Sects. 4 and 5. Throughout
this paper, we adopt a flat $\Lambda$CDM cosmology with
$H_{0}=67.7\,\text{km\,s}^{-1}\,\text{Mpc}^{-1}$ and $\Omega_{m}=0.308$
(Planck Collaboration et al. 2020). Therefore $z=0.077$ corresponds to a
luminosity distance of $D_{\text{ld}}=360\,\text{Mpc}$. All magnitudes will be
reported in the AB system (not corrected for Galactic extinction). All the
quoted uncertainties correspond to the 90% confidence level, unless specified
otherwise.

Figure 1: Long-term X-ray and UV light curves for J0456$-$20\. The coloured
regions represent the four phases: the plateau phase (${P}_{\text{X, plat}}$,
light cyan), the rapid drop phase (${P}_{\text{X, drop}}$, light blue), the
faint phase (${P}_{\text{X, faint}}$, light yellow), and the rising phase
(${P}_{\text{X, rise}}$, light green). Upper panel: Red points with error bars
are the unabsorbed rest-frame $0.2-2.0\,\text{keV}$ X-ray light curve from
eROSITA (hexagons), Swift/XRT (squares), NICER (circles), XMM-Newton (stars),
and Chandra (diamonds). The error bars indicate 90% uncertainties. The points
with downward arrows represent the $3\sigma$ flux/luminosity upper limits. The
points with blue colour are the data used to model the profiles of the
${P}_{\text{X, rise}}$ phase. The green dashed lines show the best-fitting
power-law model for the five X-ray rising phases. The grey shaded regions mark
the $1\sigma$ uncertainty of the model. Bottom panel: UV light curve from
Swift/UVOT UVM2 (red squares). The error bars mark the $1\sigma$
uncertainties. Squares with downward arrows indicate 3$\sigma$ upper limits.
The vertical lines mark the dates of the ATCA radio observations (black
dashed-dotted: non-detections; blue dashed: detections).

##  2 Observations and Data Reduction

In this paper, we analysed all observations after MJD 59720. The reader should
refer to Liu et al. (2023) for the details of data analysis before this date.

###  2.1 XMM-Newton

A pre-approved XMM-Newton target of opportunity (ToO) observation was
performed on 2022 Aug 23 (hereafter X5). In addition, two XMM-Newton
Director’s Discretionary Time (DDT) observations were performed on 2023 Feb 10
and Mar 4 (hereafter X6 and X7, respectively). J0456$-$20 was detected in all
new XMM-Newton observations.

The Observation Data Files (ODFs) for each observation were reduced using the
XMM-Newton Science Analysis System software (SAS, version 19.1, Gabriel et al.
2004) with the latest calibration files. For each observation, the SAS tasks
emchain and epchain tasks were used to generate the event lists for the
European Photon Imaging Camera (EPIC) MOS (Turner et al. 2001) and pn (Strüder
et al. 2001) detectors, respectively. High background flaring periods were
identified and filtered from the event lists. For all the EPIC images, a
circular region with a radii of $30,20,25^{\prime\prime}$ was chosen as the
source region for X5, X6, and X7, respectively. A source-free annulus region
with an inner radius of $50^{\prime\prime}$ and an outer radius of
$100^{\prime\prime}$ was chosen as the background region for all MOS
observations. The background for the pn camera was extracted from a circular
region with a radius of $60^{\prime\prime}$ centred at the same CCD read-out
column as the source position for all observations. X-ray events with pattern
$\leq 12$ for MOS and $\leq 4$ for pn were selected to extract the X-ray
spectra. We used the tasks rmfgen and arfgen to generate the response matrix
and ancillary files, respectively. The X-ray spectra were rebinned to have at
least 1 count per bin.

###  2.2 Swift observations

The XRT online data analysis tool222http://www.swift.ac.uk/user_objects (Evans
et al. 2009) was used to check whether the source was detected for each
observation. It was also used to generate the X-ray spectra for observations
in which J0456$-$20 was detected and to calculate the $3\sigma$ count rate
upper limits for non-detections. The X-ray spectra were rebinned to have at
least 1 count in each bin.

The Swift/UVOT data were reduced using the UVOT analysis pipeline provided in
HEASoft (version 6.31) with UVOT calibration version 20201215. Source counts
were extracted from a circular region with a radius of $5^{\prime\prime}$
centred at the source position. A $20^{\prime\prime}$ radius circle from a
source-free region close to the position of J0456$-$20 was chosen as the
background region. The task uvotsource was used to extract the photometry.

###  2.3 NICER observations

The NICER data were analysed using HEASoft with the NICER data analysis
software (version 10) and calibration files (version 20221001). The nicerl2
task is used to generate cleaned X-ray events. Events with overshoot higher
than 1.5 or undershoot larger than 300 were removed. The nicerl3-spec task was
then used to generate the X-ray spectra for each NICER observation. The X-ray
spectra were then rebinned to have at least one count per bin. We adopted the
SCORPEON model to generate background models for each observation. The
nicerarf and nicerrmf tasks were used to generate the response matrix and
ancillary file for each observation, respectively. The same procedures were
also adopted to re-analyse the NICER observations taken during the first two
X-ray rising phases (i.e. observation carried out during MJD 59418-59448 and
59600-59641).

###  2.4 Chandra observations

We requested Chandra DDT observations of J0456$-$20, which were performed on
2023 Mar 18, Apr 11, and April 14 with the Advanced CCD Imaging Spectrometer
(ACIS). We used the CIAO (Fruscione et al. 2006, version 4.15) software
package to reduce the Chandra data with calibration files CALDB version
4.10.4. We reprocessed the Chandra data using the CIAO script chandra_repro.
The CIAO task dmextract was used to extract the source and background spectra.
We extracted the source spectra using a circular region with a radius of
$2^{\prime\prime}$. The background spectra were extracted using an annulus
(concentric with the source) region with an inner and outer radius of
$6^{\prime\prime}$ and $20^{\prime\prime}$, respectively. The response files
were generated using the mkacisrmf and mkarf tasks. The position of the X-ray
flare measured from Chandra is $(\text{RA, Dec})=(\text{04:56:49.81},-20\degr
37\arcmin 47.98\arcsec)$ with a $2\sigma$ uncertainty of $0.54\arcsec$
(Appendix A), consistent with the centre of the host galaxy.

###  2.5 ATCA radio observations

We observed the coordinates of J0456$-$20 nine times with the Australia
Telescope Compact Array (ATCA) between 2022 Aug and 2023 Mar, in addition to
the five observations between 2021 Mar and 2022 Apr reported in Liu et al.
(2023). We observed the target during the X-ray outburst phase as this was
previously when radio emission had been detected and therefore observed the
target with the array in various configurations. In each observation, we used
the dual 5.5 GHz and 9 GHz receiver, placing the 2x2 GHz of bandwidth split
into 2048x1 MHz channels at a central frequency of 5.5 GHz and 9 GHz. Data
were reduced in the Common Astronomy Software Application (CASA, verison
5.6.3, CASA Team et al. 2022) using standard procedures, including flux and
bandpass calibration with PKS 1934-638 and phase calibration with PKS
0454-234. Additionally, we carried out one round of phase-only self-
calibration of the target field at both 5.5 and 9 GHz, with a typical solution
interval of 2 minutes, to produce a good quality image due to a bright AGN in
the field. Images of the target field were created with the CASA task tclean
and where a source was visible at the location of J0456$-$20, the flux density
was extracted with the CASA task imfit by fitting a Gaussian the size of the
synthesised beam.

Figure 2: Evolution of the recurrence time. The purple square (blue circle)
points represent $T_{\text{recur}}$ measured from the ${P}_{\text{X, rise}}$
(${P}_{\text{X, drop}}$) phases. The error bars mark the $1\sigma$
uncertainties. The upward arrow indicates the $3\sigma$ lower limit. The
estimated total mass loss from the star is also marked.

##  3 Data analysis and results

###  3.1 X-ray spectral modelling

The Xspec software (version 12.13.0, Arnaud 1996) was used to fit all X-ray
spectra using the Cash statistic (Cash 1979, Cstat in Xspec). As mentioned in
Liu et al. (2023), a power-law model (i.e. TBabs*zashift*cflux*powerlaw,
hereafter $\text{M}_{\text{pl}}$) is preferred for observations taken at
relatively high X-ray flux (i.e. the rest-frame unabsorbed 0.2–2.0 keV flux,
$f_{\text{X, soft}}$, $\gtrsim 5\times
10^{-13}\text{erg\,cm}^{-2}\,\text{s}^{-1}$). We thus first fitted all the new
X-ray spectra with the $\text{M}_{\text{pl}}$ model. The Galactic column
density is fixed at $3.3\times 10^{20}\leavevmode\nobreak\ \mathrm{cm^{-2}}$.

For NICER data, we first fitted the total X-ray spectra over the 0.25–10.0 keV
range with the backgrounds model generated using SCORPEON. We then re-fitted
the data by adding the $\text{M}_{\text{pl}}$ model. A $3\sigma$ upper limit
for $f_{\text{X, soft}}$ was estimated for observations in which the fit did
not improve significantly (i.e. $\Delta C_{\text{stat}}<11.8$) after adding
the $\text{M}_{\text{pl}}$ model. A strong oxygen K$\alpha$ line is presented
in some of the NICER data. The niscorpv22_swcxok_norm parameter in the
background model was thus left free to properly model the oxygen K$\alpha$
line in those observations. We fitted the background subtracted spectra with
the $\text{M}_{\text{pl}}$ model for the data from the other missions. The
Swift/XRT X-ray spectra were fitted over the 0.3–5.0 keV energy range, while
the 0.5–5.0 keV energy band was used for Chandra observations. For XMM-Newton
data, we jointly fitted the data from the three EPIC cameras over the 0.2–5.0
keV energy range for X5 and X7 (0.2–2.0 keV for X6). The best-fitting values
and the 90 percent uncertainties were calculated for the $f_{\text{X, soft}}$
and photon index parameters. In addition, the 68 percent uncertainties for the
$f_{\text{X, soft}}$ were also estimated for observations taken during the
${P}_{\text{X, rise}}$ phase. The $3\sigma$ upper limits of $f_{\text{X,
soft}}$ were calculated for non-detections using either the
$\text{M}_{\text{pl}}$ or a disk model (see below). The details of the fitting
results are listed in Table LABEL:tab:logxray.

Liu et al. (2023) noticed that the UV-to-X-ray SEDs of J0456$-$20 can be
described by a multi-colour disk model (TBabs*zashift*cflux*diskbb, hereafter
$\text{M}_{\text{mcd}}$) when the X-ray flux is low. Thus, the
$\text{M}_{\text{mcd}}$ model was also used to fitted the X-ray spectra at the
early stage of the ${P}_{\text{X, rise}}$ phase, that is, observations taken
between MJD 59600 and 59620 in cycle 3 and X6 in cycle 5. J0456$-$20 was
detected only on MJD 59619 (NICER, ObsID: 4595020126, hereafter N26) and on
MJD 59999 (X6). The best-fitting $T_{\text{in}}$ and $f_{\text{X, soft}}$ are
$193^{+43}_{-25}\leavevmode\nobreak\ (78_{-18}^{+24})\leavevmode\nobreak\ $eV
and $2.0^{+0.5}_{-0.3}\times 10^{-14}\leavevmode\nobreak\
(2.3_{-0.5}^{+1.2}\times 10^{-13})\leavevmode\nobreak\
\text{erg\,cm}^{-2}\,\text{s}^{-1}$ for X6 (N26), respectively. The
$\text{M}_{\text{pl}}$ model resulted in a higher $f_{\text{X, soft}}$ of
$3.3^{+1.0}_{-0.6}\times 10^{-14}\leavevmode\nobreak\ (4.6^{+1.1}_{-1.6}\times
10^{-13})\leavevmode\nobreak\ \text{erg\,cm}^{-2}\,\text{s}^{-1}$ with photon
index of $3.3^{+0.3}_{-0.4}\leavevmode\nobreak\ (5.3^{+0.7}_{-0.8})$ for X6
(N26). Both the $\text{M}_{\text{mcd}}$ and the $\text{M}_{\text{pl}}$ model
fitted the X6 and N26 spectra well, though the $\text{M}_{\text{pl}}$ model
gives a slightly better fit ($C_{\text{stat}}/\text{d.o.f}=136/144$, compared
to 152/144) for X6.

In this work, we used the results from the $\text{M}_{\text{mcd}}$ model for
X6 and N26. The best-fitting $T_{\text{in}}$ of N26 is comparable to that
obtained from the eRASS2 and Swift observations (in the range of $\sim
50-100\,$eV, Liu et al. 2023) at similar $f_{\text{X, soft}}$ (i.e. $\approx
4\times 10^{-13}\,\text{erg\,cm}^{-2}\,\text{s}^{-1}$), while $T_{\text{in}}$
is much higher during the X6 observation when J0456$-$20 was in a historically
low X-ray flux. This suggests a potential change in $T_{\text{in}}$ during the
${P}_{\text{X, rise}}$ phase. For this reason, different values of
$T_{\text{in}}$ (listed in Table LABEL:tab:logxray) were used to calculate the
$3\sigma$ flux upper limits for NICER observations before N26 during cycle 3.

###  3.2 Modelling the X-ray rising phase

The ${P}_{\text{X, rise}}$ phases during cycles 2, 3, 4 and 5 were captured by
our follow-up observations. We also assumed that J0456$-$20 is in the
${P}_{\text{X, rise}}$ phase of cycle1 during the eRASS2 observations. This is
justified by the duration of the ${P}_{\text{X, drop}}$ phase being much
shorter than the ${P}_{\text{X, rise}}$ phase, and by the spectral property of
eRASS2 (i.e. best described by the $\text{M}_{\text{mcd}}$ model) being
similar to that at the early stage of the ${P}_{\text{X, rise}}$ phase.

Table 1: Results of the X-ray rising phase modelling and estimations of $T_{\text{recur}}$. Parameters: name of the parameter in the power-law model; Cycles[1,2,5]: results by jointly fitting the ${P}_{\text{X, rise}}$ phases of cycles 1, 2, and 5; All cycles: results by jointly fitting the ${P}_{\text{X, rise}}$ of all the five cycles; $T_{\text{recur}}$ and $T_{\text{recur, drop}}$ are the recurrence time estimated using the ${P}_{\text{X, rise}}$ and ${P}_{\text{X, drop}}$ phases, respectively. Parameters | Cycles[1,2,5] | All cycles | $T_{\text{recur}}$ | $T_{\text{recur, drop}}$
---|---|---|---|---
MJD | MJD | days | days
$t_{1}$ | $59070_{-8}^{+7}$ | $59065^{+8}_{-9}$
$299^{+6}_{-6}$ | $227\pm 3$
$t_{2}$ | $59367_{-5}^{+5}$ | $59363^{+6}_{-6}$
$227^{+4}_{-3}$ | $188\pm 3$
$t_{3}$ | — | $59591^{+6}_{-5}$
$188^{+3}_{-3}$ | $178\pm 3$
$t_{4}$ | — | $59778^{+7}_{-7}$
$193^{+3}_{-3}$ | $>189$
$t_{5}$ | $59976_{-4}^{+3}$ | $59972^{+4}_{-5}$
$\log A$ | $-16.2_{-0.8}^{+0.6}$ | $-17.2^{+1.0}_{-1.2}$
$\beta$ | $2.6_{-0.3}^{+0.4}$ | $3.1^{+0.6}_{-0.5}$

We jointly fitted the five ${P}_{\text{X, rise}}$ phases with a power law
function $f_{\text{rs,i}}(t)=A*(t-t_{i})^{\beta}$, where $i=1,2,3,4,5$. We
assumed that the normalisation $A$ and power law index $\beta$ are the same
for all the five ${P}_{\text{X, rise}}$ phases. The lmfit package is adopted
to fit the data with the least squares method. To take into account upper
limits and to estimate the uncertainties of the parameters in the model, we
generated $10^{5}$ realisations of the $f_{\text{X, soft}}$ in the five
${P}_{\text{X, rise}}$ phases from Gaussian distributions for observations of
which J0456$-$20 is detected and uniform distributions with a lower limit of
zero (excluded) for non-detections. The best-fitting $f_{\text{X, soft}}$ and
the $1\sigma$ uncertainties obtained from X-ray spectral modelling of each
observation are used as the means and standard deviations of the Gaussian
distributions, respectively. The $3\sigma$ $f_{\text{X, soft}}$ upper limits
for non-detections are used as the upper limits for the uniform distributions.
We obtained $10^{5}$ values for each parameter in the model by fitting the
$10^{5}$ datasets using the least squares method. We estimated the best-
fitting values using the median values for each parameter and estimated the
$1\sigma$ confidence intervals using the 16 and 84 percentiles of fitting
results. Similarly, the recurrence time $T_{\text{recur}}$ for cycle $i$ and
$i+1$, the lower, and the upper intervals of the $1\sigma$ confidence
intervals are estimated using the median, the 16, and 84 percentile of a
sample calculated using $t_{i+1}-t_{i}$. To test whether the results could be
affected by the inclusion of flux upper limits, we applied the same procedures
to cycles 1, 2, and 5 only (i.e. cycles without upper limits in the
${P}_{\text{X, rise}}$ phases). We found the results to be consistent within
$1\sigma$ uncertainties (see Table 1). The fitting results and the estimated
$T_{\text{recur}}$ are listed in Table 1. We also calculated the recurrence
time using the ${P}_{\text{X, drop}}$ phase (Appendix B). The estimated values
of $T_{\text{recur}}$ are also listed in Table 1. It is clear from Fig. 2 that
the $T_{\text{recur}}$ of the X-ray flares in J0456$-$20 show rapid changes.
The mass loss for each cycle marked in Fig. 2 are calculated using the method
outlined in Liu et al. (2023) (see Appendix C and Table LABEL:tab:massloss).

###  3.3 Radio variability

J0456$-$20 was mostly undetected after 2022 Aug, with a detection at 9 GHz on
2022 Oct 05 and at 5.5 GHz on 2022 Oct 07 during the ${P}_{\text{X, plat}}$
phase of cycle 4 (see Fig. 1) with a significance of 7 and 9$\sigma$,
respectively. In order to improve the sensitivity, we additionally stacked the
observations from October 2-7 and detected a faint point source at the
coordinates of J0456$-$20\. The radio observations reported here indicate that
the transient radio source associated with the X-ray outbursts in 2022 Mar and
Apr has also been fading, consistent with the decrease in the peak X-ray flux
of each cycle (see Fig. 1). A summary of the ATCA radio observations of
J0456$-$20 is given in Table 4. The long-term radio light curve is shown in
Fig. 5.

##  4 Discussion

We have detected five repeating X-ray flares in J0456$-$20 using the latest
data. In addition, repeating transient radio emission has also been detected
in J0456$-$20\. These results provide further evidence that J0456$-$20 is a
repeating nuclear transient, and make it one of the most promising repeating
pTDE candidates. More importantly, our results also revealed rapid evolution
of the recurrence time $T_{\text{recur}}$, measured from the ${P}_{\text{X,
rise}}$ phases, of the X-ray flares. $T_{\text{recur}}$ decreased by more than
2 months between cycles 1 and 2. It continued to decrease by roughly
$40\leavevmode\nobreak\ $days between cycles 2 and 3. Such a rapid change is
likely terminated as suggested by an almost constant $T_{\text{recur}}$ ($\sim
190\leavevmode\nobreak\ $days) measured from the latest three cycles. Evidence
for evolution of $T_{\text{recur}}$ are also found using the values measured
from the ${P}_{\text{X, drop}}$ phases (Appendix B and Table 1). Though, the
changes in $T_{\text{recur}}$ are less dramatic than those measured from the
${P}_{\text{X, rise}}$ phases, which may attributed to the changes in the
duration of the other phases (see Table LABEL:tab:massloss). In this work,
$T_{\text{recur}}$ derived from the ${P}_{\text{X, rise}}$ phase were used.

The evolution of $T_{\text{recur}}$ has been reported only in a few repeating
pTDE candidates. For instance, ASASSN-14ko showed a decrease in the periods
with a period derivative of around -0.0026 (Payne et al. 2022; Huang et al.
2023), which is much shorter than that found in J0456$-$20 ($\lesssim-0.2$).
The X-ray flares in HLX-1 initially showed a quasi-periodic $T_{\text{recur}}$
of around 1 yr. $T_{\text{recur}}$ then increased by about 1 month in 2013
(Godet et al. 2014), and continued to increase until 2018, after which no
X-ray outbursts were detected (Webb et al. 2023). Unlike in HLX-1, we found no
evidence for an increase in $T_{\text{recur}}$ in J0456$-$20 as of now. As
suggested by Godet et al. (2014), the evolution of $T_{\text{recur}}$ can put
strong constraints on the initial mass, the mass loss, and the orbital
parameters of the star.

A repeating pTDE is favoured to explain the long-term multi-wavelength light
curve of J0456$-$20 (Liu et al. 2023). We thus performed simulations to test
if the changes in $T_{\text{recur}}$, $\Delta P$, can be explained by pTDEs.
We made a grid of hydrodynamics simulations, using the moving-mesh code AREPO
(Springel 2010; Weinberger et al. 2020; Pakmor et al. 2016), where we examined
the change in the orbital period of remnants produced in pTDEs of main-
sequence stars by BHs with a mass of
$M_{\text{BH}}=10^{5}\mathrm{M}_{\odot}$333A decrease in $T_{\text{recur}}$
requires a smaller $M_{\text{BH}}$ of around $10^{5}\mathrm{M}_{\odot}$ in our
simulations which ran with a limit range of parameter space. We noted that a
$M_{\text{BH}}$ of the order of $10^{5}\mathrm{M}_{\odot}$ is possible for
J0456$-$20 as the measured BH mass in Liu et al. (2023) has a large
uncertainty.. We considered Solar metallicity main-sequence stars with masses
of $M_{\star}=1$, 2, and 3 $\mathrm{M}_{\odot}$, and a core Hydrogen mass
fraction of 0.01 (terminal-age) and 0.3 (middle-age), evolved using the 1D
stellar evolution code MESA (Paxton et al. 2013, 2015, 2019, 2011), imported
into AREPO with 0.5M cells444We confirmed that simulations with different
resolutions between 0.25M and 1M cells yield converging $\Delta P$.. We also
considered a wide range of the pericenter distance $r_{\rm p}$, $0.1\lesssim
r_{\rm p}/r_{\rm t}\lesssim 1.2$ (tidal radius $r_{\rm
t}=(M_{\text{BH}}/M_{\star})^{1/3}R_{\star}$), encompassing scenarios from the
full disruption to no mass loss. We varied the pericenter distance for a given
$M_{\star}$. Meanwhile, we fixed the orbital period of the original orbit to
be 300 days, so the stellar orbit in each simulation has a different
eccentricity ($e\gtrsim 0.99$). The initial separation between the black hole
and the star is $5$ $r_{\rm t}$. We followed the evolution of the remnant
after the first pericenter passage of the original star, using the Helmholtz
equation of state (Timmes & Swesty 2000) until the post-disruption orbital
parameters do not evolve, which occurred when the separation between the black
hole and the remnant is $\gtrsim 5r_{\rm t}$. We verified that the total
energy in all simulations was conserved within a fractional error of $\lesssim
10^{-5}$.

Fig. 3 depicts $\Delta P$ as a function of the original stellar mass
$M_{\star}$ and the fractional mass loss $\Delta M/M_{\star}$. The general
trend is that $\Delta P$ decreases as $\Delta M/M_{\star}$ increases until the
fractional mass loss exceeds a critical value $\Delta m_{\text{c}}$, roughly
$\approx 0.7-0.8$. Above the critical mass loss, $\Delta P$ starts to increase
with $\Delta M/M_{\star}$. This trend was observed for parabolic pTDEs in Ryu
et al. (2020). However, the values of $\Delta P$ strongly depend on the
internal structure of the star (i.e. mass and age). For pTDEs of middle-age
stars, $\Delta P$ is negative (positive) when $\Delta M/M_{\star}$ is below
(above) $\Delta m_{\text{c}}$, meaning the remnants become more (less) bound
than that of the original star before the TDE. For this case, $\Delta P$ is at
most $\simeq-10$ days. However, $\Delta P$ is negative across a wider range of
$\Delta M/M_{\star}$ and can be as large as $-90$ days for pTDEs of terminal-
age stars. Most notably, the large $\Delta P$ (i.e.
$\sim-70\leavevmode\nobreak\ $days) observed in J0456$-$20 between cycles 1
and 2 can be reproduced if the original star is a $1\mathrm{M}_{\odot}$
terminal-age star and loses nearly 80-90 % of its mass. While we have not
explored the entire parameter space of pTDEs, our simulations suggest that the
observed decrease in $T_{\text{recur}}$ may be explained by a severe pTDE of a
main-sequence star near the terminal age. The inferred mass loss
($0.8-0.9\,\mathrm{M}_{\odot}$) is much higher than that estimated for cycle 1
($\sim 0.12\,\mathrm{M}_{\odot}$, Appendix C and Table LABEL:tab:massloss) of
J0456$-$20\. We note that this discrepancy can be alleviated if the radiation
efficiency in J0456$-$20 is much lower than the assumed value of 0.1. It is
important to emphasise that this simulation result does not rule out other
potential mechanisms; instead, it merely suggests partial tidal disruption is
a plausible mechanism for generating a transient like J0456$-$20.

Figure 3: Change in the orbital period $\Delta P$ (in days) for remnants in
pTDEs of main-sequence stars with mass of $M_{\star}=\text{1, 2, and
3}\leavevmode\nobreak\ \mathrm{M}_{\odot}$ in hydrodynamics simulations, as a
function of the fractional mass loss $\Delta M/M_{\star}$. The original stars
are Solar metallicity main-sequence stars with a core Hydrogen mass fraction
of $X_{\rm H}\simeq 0.01$ (Terminal-age, solid lines) and $0.3$ (Middle-age,
dashed lines). The simulations suggest that $\Delta P$ in severe pTDEs of
terminal-age main-sequence stars can be compatible with the change in $T_{\rm
recur}$ between cycles 1 and 2 ($\simeq-70$ days, grey horizontal line) of
J0456-20.

##  5 Summary

In this letter, we analysed new multi-wavelength observations for the
promising repeating pTDE candidate J0456$-$20\. We have detected five
repeating X-ray flares and repeating transient radio emission from J0456$-$20,
providing additional strong evidence that J0456$-$20 is a repeating nuclear
transient. In addition, the latest X-ray data also reveals changes in the
recurrence time $T_{\text{recur}}$ of the X-ray flare. $T_{\text{recur}}$
initially decreased rapidly from $\sim 300\leavevmode\nobreak\ $days to $\sim
230$ days, it continues to decrease by roughly $40\leavevmode\nobreak\ $days
per cycle with an indication of constant values of $\sim 190$ days in the
latest cycles. Our hydrodynamic simulations show that the large decrease in
$T_{\text{recur}}$ can be explained in the pTDE scenario, providing that the
original star is a $1\,\mathrm{M}_{\odot}$ terminal-age star with an initial
fractional mass loss of around 80-90%. Our results suggest that precise
estimations of $T_{\text{recur}}$ in repeating pTDEs can provide additional
constraints on the initial mass of the disrupted star and mass loss during
each passage. They also indicate that repeating pTDEs could be effective tools
to probe the stellar/gas dynamics around SMBHs beyond our own Galaxy.

###### Acknowledgements.

ZL is grateful to the XMM-Newton, Swift, and NICER teams for approving the
ToO/DDT requests and arranging the follow-up observations. The hydrodynamics
simulations were conducted using computational resources (and/or scientific
computing services) at the Max-Planck Computing & Data Facility. DH
acknowledges support from DLR grant FKZ 50 OR 2003. MK is supported by DFG
grant KR 3338/4-1 and DLR grant 50 OR 2307. This work was supported by the
Australian government through the Australian Research Council’s Discovery
Projects funding scheme (DP200102471).

## References

  * Arcodia et al. (2021) Arcodia, R., Merloni, A., Nandra, K., et al. 2021, Nature, 592, 704
  * Arnaud (1996) Arnaud, K. A. 1996, ASP Conf. Ser., 101, 17
  * Bortolas et al. (2023) Bortolas, E., Ryu, T., Broggi, L., & Sesana, A. 2023, MNRAS, 524, 3026
  * CASA Team et al. (2022) CASA Team, Bean, B., Bhatnagar, S., et al. 2022, PASP, 134, 114501
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939
  * Davis et al. (2011) Davis, S. W., Narayan, R., Zhu, Y., et al. 2011, ApJ, 734, 111
  * Evans et al. (2009) Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2009, MNRAS, 397, 1177
  * Evans et al. (2023) Evans, P. A., Nixon, C. J., Campana, S., et al. 2023, Nature Astronomy [arXiv:2309.02500]
  * Fruscione et al. (2006) Fruscione, A., McDowell, J. C., Allen, G. E., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, 62701V
  * Gabriel et al. (2004) Gabriel, C., Denby, M., Fyfe, D. J., et al. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 314, Astronomical Data Analysis Software and Systems (ADASS) XIII, ed. F. Ochsenbein, M. G. Allen, & D. Egret, 759
  * Giustini et al. (2020) Giustini, M., Miniutti, G., & Saxton, R. D. 2020, A&A, 636, L2
  * Godet et al. (2014) Godet, O., Lombardi, J. C., Antonini, F., et al. 2014, ApJ, 793, 105
  * Guolo et al. (2023) Guolo, M., Pasham, D. R., Zajaček, M., et al. 2023, arXiv e-prints, arXiv:2309.03011
  * Hayasaki et al. (2013) Hayasaki, K., Stone, N., & Loeb, A. 2013, MNRAS, 434, 909
  * Huang et al. (2023) Huang, S., Jiang, N., Shen, R.-F., Wang, T., & Sheng, Z. 2023, ApJ, 956, L46
  * Krolik et al. (2020) Krolik, J., Piran, T., & Ryu, T. 2020, ApJ, 904, 68
  * Liu et al. (2023) Liu, Z., Malyali, A., Krumpe, M., et al. 2023, A&A, 669, A75
  * Malyali et al. (2023) Malyali, A., Liu, Z., Rau, A., et al. 2023, MNRAS, 520, 3549
  * Miniutti et al. (2023) Miniutti, G., Giustini, M., Arcodia, R., et al. 2023, A&A, 670, A93
  * Miniutti et al. (2019) Miniutti, G., Saxton, R. D., Giustini, M., et al. 2019, Nature, 573, 381
  * Pakmor et al. (2016) Pakmor, R., Springel, V., Bauer, A., et al. 2016, MNRAS, 455, 1134
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10
  * Payne et al. (2022) Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2022, ApJ, 926, 142
  * Payne et al. (2021) Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2021, ApJ, 910, 125
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
  * Ryu et al. (2020) Ryu, T., Krolik, J., Piran, T., & Noble, S. C. 2020, ApJ, 904, 100
  * Springel (2010) Springel, V. 2010, MNRAS, 401, 791
  * Strüder et al. (2001) Strüder, L., Briel, U., Dennerl, K., et al. 2001, A&A, 365, L18
  * Timmes & Swesty (2000) Timmes, F. X. & Swesty, F. D. 2000, ApJS, 126, 501
  * Turner et al. (2001) Turner, M. J. L., Abbey, A., Arnaud, M., et al. 2001, A&A, 365, L27
  * Webb et al. (2012) Webb, N., Cseh, D., Lenc, E., et al. 2012, Science, 337, 554
  * Webb et al. (2023) Webb, N. A., Barret, D., Godet, O., et al. 2023, Astronomische Nachrichten, 344, easna.20230051
  * Weinberger et al. (2020) Weinberger, R., Springel, V., & Pakmor, R. 2020, ApJS, 248, 32
  * Wevers et al. (2023) Wevers, T., Coughlin, E. R., Pasham, D. R., et al. 2023, ApJL, 942, L33

##  Appendix A Chandra astrometric correction

We ran the wavdetect tool on the Chandra data to generate a source list for
each observation. J0456$-$20 was detected in all the three Chandra
observations. We used the source positions provided in the Legacy Survey DR10
catalogue as a reference. The overall 90 per cent absolute astrometry
uncertainty of Chandra is $\sim 1.11$
arcsec555http://cxc.cfa.harvard.edu/cal/ASPECT/celmon. We thus cross-matched
the positions of the X-ray source measured from the Chandra observations with
the source positions listed in the DR10 catalogue using a radius of 1.1
arcsec. We selected sources within 3 arcmin of the aimpoint of the telescope.
This resulted in three sources (excluding J0456$-$20, see 2) that are detected
in both X-ray and DR10 in the second Chandra observation (only two for the
other two observations). The CIAO task wcs_match and wcs_update were then used
to correct and update the aspect ratio. The residual rms scatter in the
corrected X-ray positions of the LS10 sources is 0.28 arcsec, which
corresponds to a $2\sigma$ position error of $\approx 0.54$ arcsec (assuming
Rayleigh distribution). The astrometric corrected Chandra position is
consistent with the centre of the host galaxy (see Fig. 4).

Figure 4: Updated multi-wavelength positions of J0456$-$20\. The black dot marks the position given in the Gaia EDR3. The crosses represent the positions measured from different instruments, and the circles indicate the $2\sigma$ positional uncertainties: Swift/UVOT (green, $2\sigma=0.52\arcsec$), Chandra (red, $2\sigma=0.54\arcsec$), ATCA (white, $2\sigma=1.0\arcsec$), XMM-Newton (royalblue, $2\sigma=1.6\arcsec$), and eRASS3 (orange, $2\sigma=2.0\arcsec$). Table 2: Ref1–3 are the reference sources used for performing astrometric correction for Chandra. The Separations are calculated using the astrometric corrected Chandra positions and the LS DR10 positions. | Object | Coordinates/Seperations
---|---|---
LS DR10 | J0456$-$20 | 04:56:49.80, -20°37′47.99″
Ref1 | 04:56:57.39, -20°35′12.75″
Ref2 | 04:56:58.64, -20°35′57.25″
Ref3 | 04:56:58.70, -20°37′47.65″
Chandra | J0456$-$20 | 04:56:49.81, -20°37′47.98″
Ref1 | 04:56:57.39, -20°35′13.25″
With correction | Ref2 | 04:56:58.66, -20°35′57.49″
Ref3 | 04:56:58.73, -20°37′48.58″
Chandra | J0456$-$20 | 04:56:49.82, -20°37′48.54″
Ref1 | 04:56:57.38, -20°35′12.69″
No correction | Ref2 | 04:56:58.64, -20°35′56.94″
Ref3 | 04:56:58.71, -20°37′48.03″
Seperation | J0456$-$20 | 0.07″
Ref1 | 0.18″
Ref2 | 0.32″
Ref3 | 0.41″

##  Appendix B Estimation of the recurrence time using the X-ray drop phase

The recurrence time can also be determined using the ${P}_{\text{X, drop}}$
phases. The start dates of the ${P}_{\text{X, drop}}$ phases, $t_{\text{s,
drop}}$, can be well constrained in a model-independent way due to their short
duration (i.e. $f_{\text{X, soft}}$ drops by a factor of $\gtrsim 10$ within
one week). We estimated $t_{\text{s, drop}}$ in the range of MJD
$[59285,59291]$, $[59512,59518]$, and $[59699,59706]$, and $[59878,59883]$ for
cycles 1, 2, 3, and 4, respectively. The ${P}_{\text{X, drop}}$ phase was not
observed in cycle 5 because J0456$-$20 was blocked by the Sun. However, a
lower limit of $>\text{MJD}\leavevmode\nobreak\ 60072$ can be given. We
generated $10^{5}$ realisations of $t_{\text{s, drop}}$ for each cycle in
cycles 1–4, assuming uniform distributions of $t_{\text{s, drop}}$ in the
estimated MJD ranges. $T_{\text{recur}}$ and the $1\sigma$ intervals can be
estimated using the median and the 16 and the 84 percentiles of the difference
between consecutive $t_{\text{s, drop}}$ for cycles 1–4. We calculated a
$3\sigma$ upper limit of $189\leavevmode\nobreak\ $days for $T_{\text{recur}}$
between cycles 4 and 5. The estimated $T_{\text{recur}}$ are listed in Table
1.

##  Appendix C Estimation of the mass loss

Following Liu et al. (2023), a cycle is defined as the time between the start
of two consecutive ${P}_{\text{X, rise}}$ phases. The total energy released in
each cycle was estimated by

$E_{\text{tot}}=L_{\text{fa}}\Delta t_{\text{fa}}+L_{\text{pl}}\Delta
t_{\text{pl}}+\int_{0}^{\Delta
t_{\text{rs}}}L_{\text{rs}}(t)\leavevmode\nobreak\ \text{d}t\text{,}$ (1)

where $L_{\text{rs}}$ is the bolometric luminosity during the ${P}_{\text{X,
rise}}$ phase, which is calculated using the best-fitting power law (see Sect.
3.2), that is, $L_{\text{rs}}(t)=\kappa 4\pi
D_{\text{ld}}^{2}f_{\text{rs}}(t)$, where $\kappa=L_{\text{bol}}/L_{\text{X,
res 0.2-2.0\leavevmode\nobreak\ keV}}$ is the bolometric correction factor and
$D_{\text{ld}}$ is the luminosity distance. $L_{\text{pl}}$ and
$L_{\text{fa}}$ is the average bolometric luminosity during each of the
${P}_{\text{X, plat}}$ and ${P}_{\text{X, faint}}$ phases, respectively.
$\Delta t_{\text{fa}}$, $\Delta t_{\text{pl}}$, and $\Delta t_{\text{rs}}$ are
the duration of the ${P}_{\text{X, faint}}$, ${P}_{\text{X, plat}}$, and
${P}_{\text{X, rise}}$ phases, respectively. The ${P}_{\text{X, faint}}$ phase
in cycle 5 was not covered by our follow-up observations. In this work, we
assumed $\Delta t_{\text{fa}}=70\leavevmode\nobreak\ $days for cycle 5.
$L_{\text{fa}}$ is poorly constrained, we used a conservative value of
$4.2\times 10^{41}\leavevmode\nobreak\ \text{erg\,s}^{-1}$ derived from X6
($\kappa=1.36$). Following Liu et al. (2023), we used a value of $\kappa=15$
for $L_{\text{rs}}$ and 3 for $L_{\text{pl}}$. The values of the parameters
used to calculate $E_{\text{tot}}$ for each cycle can be found in Table
LABEL:tab:massloss. To calculate the total mass loss in each cycle, we assumed
that half of the tidally disrupted debris returns to the SMBH and that the
radiation efficiency is 0.1. We estimated a total released energy (mass loss)
of 1.0 (0.12), 0.4 (0.04), 0.1 (0.01), 0.03 (0.004), and $\gtrsim 0.02\times
10^{52}\leavevmode\nobreak\ \text{erg}$ ($\gtrsim 0.003\leavevmode\nobreak\
\mathrm{M}_{\odot}$) for cycles 1, 2, 3, 4, and 5, respectively. A lower limit
is given for cycle 5 as the ${P}_{\text{X, plat}}$ phase could be longer then
the value quoted in Table LABEL:tab:massloss.

Table 3: Values of the parameters used to calculate the total release energy ($E_{\text{tot}}$) and mass loss ($\Delta M$) in each cycle. The estimated values for $E_{\text{tot}}$ and $\Delta M$ are also listed. $t_{\text{s}}$ is the start date of the cycle. $\Delta t_{\text{rs}}$, $\Delta t_{\text{pl}}$, and $\Delta t_{\text{fa}}$ are the duration for the ${P}_{\text{X, rise}}$, ${P}_{\text{X, plat}}$, and ${P}_{\text{X, faint}}$ phases, respectively. | cycle 1 | cycle 2 | cycle 3 | cycle 4 | cycle 5
---|---|---|---|---|---
$t_{\text{s}}$ (MJD) | 59065 | 59363 | 59591 | 59778 | 59972
$\Delta t_{\text{rs}}$ (day) | 100 | 84 | 51 | 39 | 38
$\Delta t_{\text{pl}}$ (day) | 120 | 65 | 57 | 61 | $\gtrsim 70$
$\Delta t_{\text{fa}}$ (day) | 61 | 71 | 68 | 85 | 70
$L_{\text{pl}}$ ($10^{44}\leavevmode\nobreak\ \text{erg\,s}^{-1}$) | 4.6 | 1.1 | 0.70 | 0.27 | 0.13
$E_{\text{tot}}$ ($10^{52}\leavevmode\nobreak\ \text{erg}$) | 1.0 | 0.36 | 0.08 | 0.03 | $\gtrsim 0.02$
$\Delta M$ ($\mathrm{M}_{\odot}$) | 0.12 | 0.04 | 0.009 | 0.004 | $\gtrsim 0.003$

##  Appendix D Details of the radio and X-ray observations

Table 4 lists the ATCA radio observations for J0456$-$20\. J0456$-$20 was
detected only in the ${P}_{\text{X, plat}}$ phase during cycles 3 and 4. The
long-term radio variability is shown in Fig. 5.

The details of the X-ray observations and the X-ray spectral fitting results
are listed in Table LABEL:tab:logxray.

Table 4: ATCA radio observations of J0456$-$20

Date | Frequency | Array | Flux Density
---|---|---|---
| (GHz) | config. | ($\mu$Jy)
2021-03-21 | 5.5 | 6D | $<78$
9 | $<66$
2021-04-26 | 5.5 | 6D | $<36$
9 | $<47$
2021-08-07 | 5.5 | EW367 | $<125$
9 | $<57$
2022-03-22 | 5 | 6A | $257\pm 11$
6 | $278\pm 11$
9 | $311\pm 12$
2022-04-06 | 2.1 | 6A | $<738$
5.5 | $61\pm 18$
9 | $103\pm 23$
17 | $117\pm 22$
21 | $<198$
2022-08-28 | 5.5 | 6D | $<53.7$
9 | $<33.6$
2022-09-21 | 5.5 | 6D | $<82.2$
9 | $<67.2$
2022-09-29 | 5.5 | 6D | $<41.1$
9 | $<34.8$
2022-10-02 | 5.5 | 6D | $<57.3$
9 | $<43.5$
2022-10-05 | 5.5 | 6D | $<46.5$
9 | $75\pm 10$
2022-10-07 | 5.5 | 6D | $47\pm 5$
9 | $<40.3$
2022-10-15 | 5.5 | H214 | $<243$
9 | $<140$
2023-03-14 | 5.5 | 750C | $<114$
9 | $<66$
2023-03-31 | 5.5 | 750C | $<76.5$
9 | $<57$
2022-10-02, | 5.5 | 6D | $35\pm 5$
2022-10-05,
2022-10-07
2022-10-02, | 9 | 6D | $28\pm 5$
2022-10-05,
2022-10-07
2023-03-14, | 5.5 | 750C | $51\pm 13$
2023-03-31
2023-03-14, | 9 | 750C | $<58$
2023-03-31
666Upper limits are reported at 3$\sigma$.

Figure 5: Evolution of the radio flux density. The error bars mark the
$1\sigma$ uncertainties. The downward arrows represent the $3\sigma$ upper
limits. The black circle shows the flux density at $5.5\leavevmode\nobreak\
$GHz measured from the stacked observation from 2023 Mar. The light cyan
regions indicate the ${P}_{\text{X, plat}}$ phase in each cycle.

Table 5: Log of X-ray observations. MJD is the mid-date of the coverage for each observation; $T_{\text{exp}}$ is the effective exposure time in units of seconds. Exposure for XMM-Newton ObsID 0862770201 is calculated from combined MOS data, while the pn exposures are given for the other XMM-Newton observations; $T_{\text{in}}/\Gamma$: the inner temperature of a multi-color disk or the photon index of the power law. A value without an uncertainty means the parameter is fixed at the given value during spectral fitting; $f_{\text{X, soft}}$ is the rest frame intrinsic $0.2-2.0\,$keV flux in units of $10^{-12}\,\text{erg\,cm}^{-2}\,\text{s}^{-1}$. The $3\sigma$ upper limits are given for observations in which the J0456$-$20 is not detected; ObsID is the observation ID for each observation; Mission: the telescopes of which the data were taken; X-ray phase marks the phases of the J0456$-$20 during the observations; Cycle: the cycle the X-ray flare during which the data were taken. Quoted uncertainties are at the 90% confidence level. Upper limits are reported at $3\sigma$ MJD | $T_{\text{exp}}$ | $T_{\text{in}}/\Gamma$ | $f_{\text{X, soft}}$ | ObsID | Mission | X-ray phase | cycle
---|---|---|---|---|---|---|---
58919.15 | 325 | 3.0 | $<0.22$ | eRASS1 | eROSITA | — | —
59101.73 | 340 | $64_{-18}^{+29}$ | $0.51_{-0.26}^{+0.51}$ | eRASS2${}^{\text{a}}$ | eROSITA | ${P}_{\text{X, rise}}$ | 1
59272.63 | 194 | $2.5_{-0.2}^{+0.2}$ | $7.76_{-0.76}^{+0.84}$ | eRASS3 | eROSITA | ${P}_{\text{X, plat}}$ | 1
59281.86 | 123 | $2.8_{-0.2}^{+0.2}$ | $11.30_{-1.40}^{+1.61}$ | eRASS3 | eROSITA | ${P}_{\text{X, plat}}$ | 1
59284.89 | 1683 | $2.5_{-0.1}^{+0.1}$ | $11.64_{-1.45}^{+1.66}$ | 00014135001 | Swift | ${P}_{\text{X, plat}}$ | 1
59290.94 | 1664 | 3.0 | $<0.19$ | 4604010101 | NICER | ${P}_{\text{X, drop}}$ | 1
59291.53 | 9941 | 3.0 | $<0.13$ | 4604010102 | NICER | ${P}_{\text{X, drop}}$ | 1
59292.21 | 6394 | 3.0 | $<0.14$ | 4604010103 | NICER | ${P}_{\text{X, drop}}$ | 1
59298.15 | 4387 | 3.0 | $<0.18$ | 4604010201 | NICER | ${P}_{\text{X, drop}}$ | 1
59300.20 | 10040 | $99_{-40}^{+120}$ | $0.03_{-0.02}^{+0.06}$ | 0862770201 | XMM-Newton | ${P}_{\text{X, drop}}$ | 1
59305.09 | 2735 | 3.0 | $<0.20$ | 4604010301 | NICER | ${P}_{\text{X, faint}}$ | 1
59320.20 | 3254 | 3.0 | $<0.28$ | 00014135002 | Swift | ${P}_{\text{X, faint}}$ | 1
59322.59 | 1808 | 3.0 | $<0.30$ | 4604010501 | NICER | ${P}_{\text{X, faint}}$ | 1
59323.49 | 4525 | 3.0 | $<0.21$ | 4604010502 | NICER | ${P}_{\text{X, faint}}$ | 1
59325.27 | 230 | 3.0 | $<0.35$ | 4604010504 | NICER | ${P}_{\text{X, faint}}$ | 1
59328.31 | 420 | 3.0 | $<0.24$ | 4604010507 | NICER | ${P}_{\text{X, faint}}$ | 1
59329.73 | 3342 | 3.0 | $<0.24$ | 00014135003 | Swift | ${P}_{\text{X, faint}}$ | 1
59340.45 | 1402 | 3.0 | $<0.54$ | 00014135004 | Swift | ${P}_{\text{X, faint}}$ | 1
59391.16 | 7669 | 78 | $0.18_{-0.11}^{+0.18}$ | 00014135005${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59399.00 | 6406 | $4.3_{-1.1}^{+1.3}$ | $0.47_{-0.27}^{+0.70}$ | 00014135006${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59405.13 | 1793 | $5.1_{-1.6}^{+1.9}$ | $1.65_{-1.14}^{+4.34}$ | 00014135007${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59413.03 | 1641 | $2.6_{-0.7}^{+0.7}$ | $0.68_{-0.29}^{+0.53}$ | 00014135008${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59422.51 | 4304 | $3.5_{-0.1}^{+0.1}$ | $2.45_{-0.16}^{+0.21}$ | 4595020102${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59423.35 | 2589 | $3.5_{-0.1}^{+0.1}$ | $2.45_{-0.16}^{+0.21}$ | 4595020103${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59427.10 | 1631 | $2.9_{-0.5}^{+0.6}$ | $1.21_{-0.45}^{+0.74}$ | 00014135009${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59428.95 | 788 | $3.6_{-0.5}^{+0.2}$ | $3.05_{-0.91}^{+0.84}$ | 4595020105${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59432.96 | 764 | $3.7_{-0.2}^{+0.3}$ | $3.37_{-0.64}^{+1.00}$ | 4595020107${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59433.12 | 999 | $3.7_{-0.2}^{+0.3}$ | $3.37_{-0.64}^{+1.00}$ | 4595020108${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59434.71 | 1968 | $3.5_{-0.5}^{+0.5}$ | $3.07_{-0.96}^{+1.45}$ | 00014135010${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 2
59441.95 | 514 | $2.9_{-0.1}^{+0.1}$ | $4.55_{-0.32}^{+0.28}$ | 4595020109${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59442.05 | 224 | $2.9_{-0.1}^{+0.1}$ | $4.55_{-0.32}^{+0.28}$ | 4595020110${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59445.50 | 2121 | $3.2_{-0.1}^{+0.1}$ | $6.15_{-0.18}^{+0.19}$ | 4595020111${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 2
59447.51 | 40200 | — | $4.79_{-0.90}^{+1.10}$ | 0891801101 | XMM-Newton | ${P}_{\text{X, plat}}$ | 2
59448.38 | 1179 | $3.2_{-0.1}^{+0.1}$ | $5.06_{-0.21}^{+0.23}$ | 4595020112 | NICER | ${P}_{\text{X, plat}}$ | 2
59450.80 | 3350 | $2.9_{-0.0}^{+0.0}$ | $4.48_{-0.13}^{+0.13}$ | 4595020113 | NICER | ${P}_{\text{X, plat}}$ | 2
59453.89 | 2954 | $2.8_{-0.2}^{+0.2}$ | $3.20_{-0.59}^{+0.72}$ | 00014135011 | Swift | ${P}_{\text{X, plat}}$ | 2
59460.31 | 345 | $3.4_{-0.2}^{+0.2}$ | $5.26_{-0.67}^{+0.76}$ | eRASS4 | eROSITA | ${P}_{\text{X, plat}}$ | 2
59467.45 | 1568 | $2.8_{-0.3}^{+0.3}$ | $2.39_{-0.29}^{+0.42}$ | 4604010901 | NICER | ${P}_{\text{X, plat}}$ | 2
59468.45 | 445 | $2.9_{-0.9}^{+0.9}$ | $1.73_{-0.91}^{+2.05}$ | 00014135012 | Swift | ${P}_{\text{X, plat}}$ | 2
59470.50 | 341 | $4.1_{-0.9}^{+1.1}$ | $1.67_{-0.61}^{+1.21}$ | 4604010902 | NICER | ${P}_{\text{X, plat}}$ | 2
59472.51 | 2542 | $2.6_{-0.4}^{+0.4}$ | $1.71_{-0.44}^{+0.61}$ | 00014135013 | Swift | ${P}_{\text{X, plat}}$ | 2
59476.22 | 1560 | $2.9_{-0.4}^{+0.4}$ | $1.23_{-0.17}^{+0.24}$ | 4604010903 | NICER | ${P}_{\text{X, plat}}$ | 2
59479.33 | 929 | $3.0_{-0.3}^{+0.3}$ | $2.81_{-0.35}^{+0.49}$ | 4604010904 | NICER | ${P}_{\text{X, plat}}$ | 2
59481.30 | 85960 | — | $2.40_{-0.36}^{+0.76}$ | 0891801701 | XMM-Newton | ${P}_{\text{X, plat}}$ | 2
59481.49 | 2490 | $2.7_{-0.4}^{+0.4}$ | $1.48_{-0.41}^{+0.58}$ | 00014135014 | Swift | ${P}_{\text{X, plat}}$ | 2
59482.46 | 4637 | $2.9_{-0.3}^{+0.3}$ | $2.33_{-0.26}^{+0.37}$ | 4604010905 | NICER | ${P}_{\text{X, plat}}$ | 2
59485.04 | 1183 | $2.7_{-0.3}^{+0.3}$ | $2.03_{-0.23}^{+0.31}$ | 4604010906 | NICER | ${P}_{\text{X, plat}}$ | 2
59488.08 | 1532 | $3.0_{-0.3}^{+0.3}$ | $2.78_{-0.38}^{+0.57}$ | 4604010907 | NICER | ${P}_{\text{X, plat}}$ | 2
59491.05 | 1410 | $2.7_{-0.3}^{+0.3}$ | $2.94_{-0.34}^{+0.49}$ | 4604010908 | NICER | ${P}_{\text{X, plat}}$ | 2
59494.25 | 811 | $2.5_{-0.3}^{+0.3}$ | $2.34_{-0.24}^{+0.31}$ | 4604010909 | NICER | ${P}_{\text{X, plat}}$ | 2
59496.67 | 3079 | $2.7_{-0.3}^{+0.3}$ | $2.05_{-0.49}^{+0.64}$ | 00014135016 | Swift | ${P}_{\text{X, plat}}$ | 2
59497.25 | 1133 | $2.9_{-0.3}^{+0.3}$ | $2.63_{-0.31}^{+0.44}$ | 4604010910 | NICER | ${P}_{\text{X, plat}}$ | 2
59499.80 | 1703 | $2.5_{-0.3}^{+0.3}$ | $2.19_{-0.24}^{+0.31}$ | 4604010911 | NICER | ${P}_{\text{X, plat}}$ | 2
59500.64 | 3647 | $2.9_{-0.3}^{+0.3}$ | $2.47_{-0.28}^{+0.41}$ | 4604010912 | NICER | ${P}_{\text{X, plat}}$ | 2
59501.35 | 10186 | $2.9_{-0.2}^{+0.2}$ | $2.13_{-0.22}^{+0.31}$ | 4604010913 | NICER | ${P}_{\text{X, plat}}$ | 2
59502.49 | 7885 | $2.7_{-0.3}^{+0.3}$ | $1.96_{-0.18}^{+0.24}$ | 4604010914 | NICER | ${P}_{\text{X, plat}}$ | 2
59503.48 | 7906 | $2.9_{-0.2}^{+0.2}$ | $2.48_{-0.24}^{+0.34}$ | 4604010915 | NICER | ${P}_{\text{X, plat}}$ | 2
59504.51 | 4370 | $2.8_{-0.3}^{+0.3}$ | $2.23_{-0.24}^{+0.35}$ | 4604010916 | NICER | ${P}_{\text{X, plat}}$ | 2
59505.52 | 3600 | $2.7_{-0.3}^{+0.3}$ | $2.36_{-0.25}^{+0.35}$ | 4604010917 | NICER | ${P}_{\text{X, plat}}$ | 2
59506.22 | 826 | $2.8_{-0.3}^{+0.3}$ | $2.41_{-0.29}^{+0.40}$ | 4604010918 | NICER | ${P}_{\text{X, plat}}$ | 2
59508.15 | 553 | $3.3_{-0.5}^{+0.5}$ | $1.95_{-0.39}^{+0.60}$ | 4604010920 | NICER | ${P}_{\text{X, plat}}$ | 2
59510.21 | 3079 | $2.8_{-0.3}^{+0.3}$ | $2.31_{-0.26}^{+0.38}$ | 4604010921 | NICER | ${P}_{\text{X, plat}}$ | 2
59510.83 | 2777 | $2.3_{-0.3}^{+0.3}$ | $1.71_{-0.39}^{+0.51}$ | 00014135017 | Swift | ${P}_{\text{X, plat}}$ | 2
59511.91 | 2492 | $2.9_{-0.3}^{+0.3}$ | $1.91_{-0.20}^{+0.27}$ | 4604010922 | NICER | ${P}_{\text{X, plat}}$ | 2
59517.60 | 2745 | 3.0 | $<0.15$ | 4604011001 | NICER | ${P}_{\text{X, drop}}$ | 2
59520.47 | 3000 | 3.0 | $<0.22$ | 4604011002 | NICER | ${P}_{\text{X, faint}}$ | 2
59523.54 | 1200 | 3.0 | $<0.27$ | 4604011003 | NICER | ${P}_{\text{X, faint}}$ | 2
59524.53 | 1371 | 3.0 | $<1.50$ | 00014135018 | Swift | ${P}_{\text{X, faint}}$ | 2
59526.63 | 936 | 3.0 | $<0.29$ | 4604011004 | NICER | ${P}_{\text{X, faint}}$ | 2
59527.63 | 3924 | 3.0 | $<0.24$ | 00014135019 | Swift | ${P}_{\text{X, faint}}$ | 2
59529.83 | 636 | 3.0 | $<0.30$ | 4604011005 | NICER | ${P}_{\text{X, faint}}$ | 2
59529.87 | 1402 | 3.0 | $<0.69$ | 00014135021 | Swift | ${P}_{\text{X, faint}}$ | 2
59532.54 | 1143 | 3.0 | $<0.29$ | 4604011006 | NICER | ${P}_{\text{X, faint}}$ | 2
59535.32 | 1817 | 3.0 | $<0.28$ | 4604011007 | NICER | ${P}_{\text{X, faint}}$ | 2
59538.23 | 2041 | 3.0 | $<0.19$ | 4604011008 | NICER | ${P}_{\text{X, faint}}$ | 2
59541.33 | 1884 | 3.0 | $<0.14$ | 4604011009 | NICER | ${P}_{\text{X, faint}}$ | 2
59543.75 | 1996 | 3.0 | $<0.39$ | 00014135022 | Swift | ${P}_{\text{X, faint}}$ | 2
59557.41 | 2372 | 3.0 | $<0.41$ | 00014135023 | Swift | ${P}_{\text{X, faint}}$ | 2
59562.50 | 193 | 3.0 | $<3.60$ | 00014135024 | Swift | ${P}_{\text{X, faint}}$ | 2
59566.74 | 614 | 3.0 | $<1.60$ | 00014135025 | Swift | ${P}_{\text{X, faint}}$ | 2
59572.58 | 2635 | 3.0 | $<0.38$ | 00014135026 | Swift | ${P}_{\text{X, faint}}$ | 2
59584.90 | 2886 | 3.0 | $<0.39$ | 00014135027 | Swift | ${P}_{\text{X, faint}}$ | 2
59601.40 | 819 | 180 | $<0.27$ | 4595020114 | NICER | ${P}_{\text{X, rise}}$ | 3
59602.56 | 3132 | 180 | $<0.05$ | 4595020115${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59603.21 | 1684 | 180 | $<0.11$ | 4595020116 | NICER | ${P}_{\text{X, rise}}$ | 3
59604.76 | 432 | 120 | $<0.21$ | 4595020117 | NICER | ${P}_{\text{X, rise}}$ | 3
59605.47 | 425 | 120 | $<0.37$ | 4595020118 | NICER | ${P}_{\text{X, rise}}$ | 3
59607.27 | 765 | 120 | $<0.19$ | 4595020119 | NICER | ${P}_{\text{X, rise}}$ | 3
59608.60 | 2918 | 120 | $<0.08$ | 4595020120 | NICER | ${P}_{\text{X, rise}}$ | 3
59609.72 | 1146 | 80 | $<0.23$ | 4595020121 | NICER | ${P}_{\text{X, rise}}$ | 3
59610.47 | 2152 | 80 | $<0.13$ | 4595020122 | NICER | ${P}_{\text{X, rise}}$ | 3
59611.79 | 754 | 80 | $<0.29$ | 4595020123 | NICER | ${P}_{\text{X, rise}}$ | 3
59612.47 | 758 | 80 | $<0.08$ | 4595020124${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59613.37 | 1443 | 80 | $<0.19$ | 4595020125 | NICER | ${P}_{\text{X, rise}}$ | 3
59619.47 | 5808 | $78_{-18}^{+24}$ | $0.23_{-0.05}^{+0.12}$ | 4595020126${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59625.99 | 787 | $3.5_{-0.2}^{+0.3}$ | $0.67_{-0.09}^{+0.11}$ | 4595020127${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59626.06 | 1584 | $3.5_{-0.2}^{+0.3}$ | $0.67_{-0.09}^{+0.11}$ | 4595020128${}^{\text{a,b}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59633.19 | 3669 | $3.6_{-0.2}^{+0.2}$ | $1.11_{-0.11}^{+0.11}$ | 4595020129${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59635.13 | 10129 | $3.2_{-0.3}^{+0.3}$ | $0.77_{-0.20}^{+0.27}$ | 00014135099${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 3
59640.03 | 3707 | $3.1_{-0.1}^{+0.1}$ | $1.43_{-0.10}^{+0.10}$ | 4604010104${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59640.03 | 3707 | $3.1_{-0.1}^{+0.1}$ | $1.43_{-0.10}^{+0.10}$ | 4604010105${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 3
59642.13 | 4069 | $3.2_{-0.1}^{+0.1}$ | $1.40_{-0.13}^{+0.08}$ | 4604010106 | NICER | ${P}_{\text{X, plat}}$ | 3
59643.68 | 2485 | $3.2_{-0.5}^{+0.5}$ | $1.78_{-0.58}^{+0.88}$ | 00014135033 | Swift | ${P}_{\text{X, plat}}$ | 3
59645.48 | 2460 | $3.2_{-0.3}^{+0.3}$ | $1.71_{-0.26}^{+0.38}$ | 4604010107 | NICER | ${P}_{\text{X, plat}}$ | 3
59648.20 | 1991 | $3.2_{-0.4}^{+0.4}$ | $1.54_{-0.24}^{+0.36}$ | 4604010108 | NICER | ${P}_{\text{X, plat}}$ | 3
59650.19 | 2977 | $3.2_{-0.5}^{+0.5}$ | $1.58_{-0.56}^{+0.88}$ | 00014135034 | Swift | ${P}_{\text{X, plat}}$ | 3
59651.90 | 668 | $2.6_{-0.6}^{+0.6}$ | $1.39_{-0.24}^{+0.35}$ | 4604010109 | NICER | ${P}_{\text{X, plat}}$ | 3
59652.06 | 720 | $3.6_{-0.5}^{+0.6}$ | $2.12_{-0.52}^{+0.91}$ | 4604010110 | NICER | ${P}_{\text{X, plat}}$ | 3
59654.83 | 1787 | $3.7_{-0.4}^{+0.4}$ | $1.83_{-0.37}^{+0.59}$ | 4604010111 | NICER | ${P}_{\text{X, plat}}$ | 3
59656.16 | 2665 | $3.0_{-0.4}^{+0.4}$ | $1.48_{-0.47}^{+0.68}$ | 00014135036 | Swift | ${P}_{\text{X, plat}}$ | 3
59657.15 | — | — | $1.92_{-0.04}^{+0.04}$ | 0884960601 | XMM-Newton | ${P}_{\text{X, plat}}$ | 3
59663.82 | 3172 | $2.8_{-0.4}^{+0.4}$ | $1.52_{-0.42}^{+0.58}$ | 00014135037 | Swift | ${P}_{\text{X, plat}}$ | 3
59670.56 | 3127 | $2.5_{-0.4}^{+0.4}$ | $1.22_{-0.32}^{+0.44}$ | 00014135038 | Swift | ${P}_{\text{X, plat}}$ | 3
59678.08 | 4305 | $2.8_{-0.4}^{+0.5}$ | $0.68_{-0.22}^{+0.34}$ | 00014135039 | Swift | ${P}_{\text{X, plat}}$ | 3
59684.37 | 2922 | $2.6_{-0.5}^{+0.5}$ | $0.88_{-0.29}^{+0.44}$ | 00014135040 | Swift | ${P}_{\text{X, plat}}$ | 3
59691.98 | 6391 | $3.1_{-0.3}^{+0.4}$ | $1.08_{-0.28}^{+0.39}$ | 00014135041 | Swift | ${P}_{\text{X, plat}}$ | 3
59698.54 | 2627 | $3.5_{-0.8}^{+0.8}$ | $1.18_{-0.56}^{+1.06}$ | 00014135042 | Swift | ${P}_{\text{X, plat}}$ | 3
59705.30 | 3014 | $3.0_{-3.0}^{+-3.0}$ | $0.12_{-0.08}^{+0.13}$ | 00014135043 | Swift | ${P}_{\text{X, drop}}$ | 3
59752.82 | 1459 | 3.0 | $<0.50$ | 00014135044 | Swift | ${P}_{\text{X, faint}}$ | 3
59756.64 | 2618 | 3.0 | $<0.28$ | 00014135045 | Swift | ${P}_{\text{X, faint}}$ | 3
59759.19 | 3092 | 3.0 | $<0.25$ | 00014135046 | Swift | ${P}_{\text{X, faint}}$ | 3
59766.09 | 2668 | 3.0 | $<0.31$ | 00014135047 | Swift | ${P}_{\text{X, faint}}$ | 3
59773.49 | 2680 | 3.0 | $<0.43$ | 00014135048 | Swift | ${P}_{\text{X, faint}}$ | 3
59780.34 | 3377 | 3.0 | $<0.25$ | 00014135049 | Swift | ${P}_{\text{X, rise}}$ | 4
59787.41 | 2976 | 3.0 | $<0.40$ | 00014135050 | Swift | ${P}_{\text{X, rise}}$ | 4
59794.63 | 2720 | 3.0 | $<0.51$ | 00014135051 | Swift | ${P}_{\text{X, rise}}$ | 4
59801.23 | 4023 | $3.1_{-2.2}^{+2.2}$ | $0.12_{-0.09}^{+0.46}$ | 00014135052${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 4
59806.58 | 2627 | $3.0_{-102.0}^{+-102.0}$ | $<0.15$ | 5202950108${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 4
59811.17 | 187 | $3.5_{-0.9}^{+1.3}$ | $0.77_{-0.39}^{+0.41}$ | 5202950109${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 4
59811.48 | 3179 | $2.3_{-0.9}^{+0.9}$ | $0.24_{-0.12}^{+0.24}$ | 00014135053${}^{\text{a}}$ | Swift | ${P}_{\text{X, rise}}$ | 4
59813.97 | 366 | $3.4_{-0.7}^{+0.8}$ | $0.55_{-0.24}^{+0.27}$ | 5202950110${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 4
59814.03 | 256 | $2.8_{-0.6}^{+0.7}$ | $0.53_{-0.18}^{+0.23}$ | 5202950111${}^{\text{a}}$ | NICER | ${P}_{\text{X, rise}}$ | 4
59814.12 | 9662 | $3.17_{-0.06}^{+0.06}$ | $0.70_{-0.03}^{+0.03}$ | 0884962601${}^{\text{a}}$ | XMM-Newton | ${P}_{\text{X, rise}}$ | 4
59819.86 | 2570 | $2.4_{-0.6}^{+0.7}$ | $0.43_{-0.19}^{+0.33}$ | 00014135455 | Swift | ${P}_{\text{X, plat}}$ | 4
59825.17 | 3384 | $2.7_{-0.7}^{+0.7}$ | $0.38_{-0.18}^{+0.33}$ | 00014135056 | Swift | ${P}_{\text{X, plat}}$ | 4
59828.19 | 3303 | $4.0_{-0.3}^{+0.4}$ | $0.78_{-0.10}^{+0.17}$ | 5202950114 | NICER | ${P}_{\text{X, plat}}$ | 4
59832.46 | 3117 | $2.7_{-0.5}^{+0.5}$ | $0.68_{-0.24}^{+0.38}$ | 00014135057 | Swift | ${P}_{\text{X, plat}}$ | 4
59839.22 | 2647 | $2.9_{-0.7}^{+0.7}$ | $0.58_{-0.26}^{+0.49}$ | 00014135058 | Swift | ${P}_{\text{X, plat}}$ | 4
59839.53 | 2717 | $3.0_{-0.2}^{+0.2}$ | $0.63_{-0.07}^{+0.07}$ | 5202950115 | NICER | ${P}_{\text{X, plat}}$ | 4
59842.32 | 1380 | $4.4_{-1.0}^{+1.8}$ | $0.64_{-0.25}^{+0.42}$ | 5202950116 | NICER | ${P}_{\text{X, plat}}$ | 4
59845.39 | 1529 | $3.6_{-0.5}^{+0.5}$ | $0.62_{-0.18}^{+0.14}$ | 5202950117 | NICER | ${P}_{\text{X, plat}}$ | 4
59846.51 | 2605 | $3.4_{-1.2}^{+1.2}$ | $0.56_{-0.31}^{+0.82}$ | 00014135059 | Swift | ${P}_{\text{X, plat}}$ | 4
59847.94 | 841 | $3.1_{-0.5}^{+0.6}$ | $0.57_{-0.18}^{+0.19}$ | 5202950118 | NICER | ${P}_{\text{X, plat}}$ | 4
59848.10 | 2940 | $2.8_{-0.3}^{+0.3}$ | $0.65_{-0.20}^{+0.09}$ | 5202950119 | NICER | ${P}_{\text{X, plat}}$ | 4
59851.44 | 1555 | $3.8_{-0.4}^{+0.4}$ | $0.81_{-0.15}^{+0.20}$ | 5202950120 | NICER | ${P}_{\text{X, plat}}$ | 4
59852.96 | 341 | $3.0$ | $<0.73$ | 5202950121 | NICER | ${P}_{\text{X, plat}}$ | 4
59853.86 | 2640 | $3.3_{-0.8}^{+0.9}$ | $0.67_{-0.35}^{+0.77}$ | 00014135060 | Swift | ${P}_{\text{X, plat}}$ | 4
59857.19 | 2144 | $3.7_{-0.3}^{+0.5}$ | $0.75_{-0.13}^{+0.20}$ | 5202950123 | NICER | ${P}_{\text{X, plat}}$ | 4
59860.20 | 1766 | $3.8_{-0.3}^{+0.5}$ | $0.71_{-0.12}^{+0.19}$ | 5202950124 | NICER | ${P}_{\text{X, plat}}$ | 4
59860.82 | 2555 | $3.4_{-0.7}^{+0.8}$ | $0.82_{-0.39}^{+0.79}$ | 00014135061 | Swift | ${P}_{\text{X, plat}}$ | 4
59863.58 | 4048 | $3.3_{-0.3}^{+0.2}$ | $0.57_{-0.10}^{+0.06}$ | 5202950125 | NICER | ${P}_{\text{X, plat}}$ | 4
59867.30 | 2122 | $3.5_{-0.5}^{+0.8}$ | $0.57_{-0.17}^{+0.10}$ | 5202950127 | NICER | ${P}_{\text{X, plat}}$ | 4
59869.51 | 977 | $4.1_{-0.5}^{+0.6}$ | $0.83_{-0.20}^{+0.29}$ | 5202950128 | NICER | ${P}_{\text{X, plat}}$ | 4
59872.21 | 392 | $3.0$ | $<0.56$ | 5202950129 | NICER | ${P}_{\text{X, plat}}$ | 4
59874.88 | 2650 | $2.8_{-0.7}^{+0.8}$ | $0.42_{-0.20}^{+0.39}$ | 00014135062 | Swift | ${P}_{\text{X, plat}}$ | 4
59874.91 | 2416 | $2.6_{-0.3}^{+0.3}$ | $0.65_{-0.11}^{+0.17}$ | 5202950130 | NICER | ${P}_{\text{X, plat}}$ | 4
59875.49 | 316 | 3.0 | $<0.33$ | 5202950131 | NICER | ${P}_{\text{X, plat}}$ | 4
59875.49 | 1211 | $3.4_{-1.0}^{+1.4}$ | $0.24_{-0.15}^{+0.15}$ | 5202950131 | NICER | ${P}_{\text{X, plat}}$ | 4
59876.52 | 1326 | $3.9_{-0.6}^{+0.8}$ | $0.44_{-0.12}^{+0.23}$ | 5202950132 | NICER | ${P}_{\text{X, plat}}$ | 4
59876.52 | 459 | $3.7_{-0.9}^{+1.2}$ | $0.49_{-0.18}^{+0.29}$ | 5202950132 | NICER | ${P}_{\text{X, plat}}$ | 4
59877.30 | 2150 | $3.0_{-0.2}^{+0.3}$ | $0.47_{-0.07}^{+0.08}$ | 5202950133 | NICER | ${P}_{\text{X, plat}}$ | 4
59877.30 | 1266 | $3.1_{-0.4}^{+0.5}$ | $0.53_{-0.10}^{+0.13}$ | 5202950133 | NICER | ${P}_{\text{X, plat}}$ | 4
59878.36 | 1955 | $4.3_{-0.5}^{+0.6}$ | $0.49_{-0.12}^{+0.17}$ | 5202950134 | NICER | ${P}_{\text{X, drop}}$ | 4
59879.59 | 3164 | $3.0$ | $<0.55$ | 5202950135 | NICER | ${P}_{\text{X, drop}}$ | 4
59879.59 | 1564 | $4.1_{-0.6}^{+0.7}$ | $0.43_{-0.12}^{+0.16}$ | 5202950135 | NICER | ${P}_{\text{X, drop}}$ | 4
59881.55 | 1119 | $2.3_{-0.7}^{+0.7}$ | $0.33_{-0.07}^{+0.08}$ | 5202950136 | NICER | ${P}_{\text{X, drop}}$ | 4
59881.55 | 3311 | $1.4_{-1.4}^{+0.6}$ | $0.17_{-0.09}^{+0.06}$ | 5202950136 | NICER | ${P}_{\text{X, drop}}$ | 4
59882.80 | 8366 | $3.0_{-3.0}^{+-3.0}$ | $0.07_{-0.04}^{+0.06}$ | 14136364 | Swift | ${P}_{\text{X, drop}}$ | 4
59884.59 | 3046 | $3.0_{-102.0}^{+-102.0}$ | $<0.11$ | 5202950137 | NICER | ${P}_{\text{X, drop}}$ | 4
59884.59 | 2552 | 3.0 | $<0.14$ | 5202950137 | NICER | ${P}_{\text{X, drop}}$ | 4
59887.94 | 2868 | 3.0 | $<0.36$ | 00014135065 | Swift | ${P}_{\text{X, faint}}$ | 4
59895.56 | 2688 | 3.0 | $<0.35$ | 00014135066 | Swift | ${P}_{\text{X, faint}}$ | 4
59920.62 | 3455 | 3.0 | $<0.31$ | 00014135067 | Swift | ${P}_{\text{X, faint}}$ | 4
59941.75 | 4054 | 3.0 | $<0.32$ | 00014135068 | Swift | ${P}_{\text{X, faint}}$ | 4
59962.62 | 3984 | 3.0 | $<0.23$ | 00014135069 | Swift | ${P}_{\text{X, faint}}$ | 4
59980.96 | 2963 | 200 | $<0.25$ | 00014135070 | Swift | ${P}_{\text{X, rise}}$ | 5
59985.53 | 15381 | $193_{-25}^{+43}$ | $0.021_{-0.003}^{+0.005}$ | 0915390401${}^{\text{a}}$ | XMM-Newton | ${P}_{\text{X, rise}}$ | 5
59990.59 | 2926 | 100 | $<0.67$ | 00014135071 | Swift | ${P}_{\text{X, rise}}$ | 5
59999.79 | 3384 | $3.4_{-2.3}^{+2.4}$ | $0.26_{-0.17}^{+0.89}$ | 00014135072 | Swift | ${P}_{\text{X, rise}}$ | 5
60007.25 | 11300 | $3.06_{-0.07}^{+0.07}$ | $0.44_{-0.02}^{+0.03}$ | 0915390501${}^{\text{a}}$ | XMM-Newton | ${P}_{\text{X, rise}}$ | 5
60009.45 | 2690 | $3.2_{-1.3}^{+1.3}$ | $0.53_{-0.27}^{+0.75}$ | 00014135073 | Swift | ${P}_{\text{X, rise}}$ | 5
60021.39 | 3189 | $2.2_{-0.9}^{+0.9}$ | $0.27_{-0.13}^{+0.25}$ | 00014135074 | Swift | ${P}_{\text{X, plat}}$ | 5
60023.00 | 9948 | $2.6_{-0.4}^{+0.4}$ | $0.29_{-0.12}^{+0.24}$ | 27737 | Chandra | ${P}_{\text{X, plat}}$ | 5
60031.16 | 2680 | $3.2_{-0.8}^{+0.8}$ | $0.63_{-0.30}^{+0.58}$ | 00014135075 | Swift | ${P}_{\text{X, plat}}$ | 5
60045.09 | 10059 | $2.6_{-0.5}^{+0.5}$ | $0.25_{-0.12}^{+0.27}$ | 27738 | Chandra | ${P}_{\text{X, plat}}$ | 5
60048.22 | 9948 | $2.5_{-0.5}^{+0.5}$ | $0.19_{-0.09}^{+0.20}$ | 27798 | Chandra | ${P}_{\text{X, plat}}$ | 5
60072.18 | 4795 | $3.0_{-0.8}^{+0.9}$ | $0.28_{-0.15}^{+0.32}$ | 00014135076 | Swift | ${P}_{\text{X, plat}}$ | 5
Table 5: continued. 777a: observations used in modelling the profile of the
${P}_{\text{X, rise}}$ phase.
b: the X-ray spectral fitting results were from jointly fitting of the two
consecutive observations.


* [2] Stephen L. Adler.  Einstein Gravity as a Symmetry Breaking Effect in Quantum Field Theory.  Rev. Mod. Phys., 54:729, 1982.  [Erratum: Rev. Mod. Phys.55,837(1983)].
  * [3] S. Weinberg.  Ultraviolet Divergences In Quantum Theories Of Gravitation.  In Hawking, S.W., Israel, W.: General Relativity; Cambridge University Press, pages 790–831, 1980.
  * [4] John F. Donoghue.  General relativity as an effective field theory: The leading quantum corrections.  Phys. Rev., D50:3874–3888, 1994, gr-qc/9405057.
  * [5] M. Reuter.  Nonperturbative evolution equation for quantum gravity.  Phys.Rev., D57:971–985, 1998, hep-th/9605030.
  * [6] Robert Percacci.  An Introduction to Covariant Quantum Gravity and Asymptotic Safety, volume 3 of 100 Years of General Relativity.  World Scientific, 2017.
  * [7] Martin Reuter and Frank Saueressig.  Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic Safety.  Cambridge University Press, 1 2019.
  * [8] Alfio Bonanno, Astrid Eichhorn, Holger Gies, Jan M. Pawlowski, Roberto Percacci, Martin Reuter, Frank Saueressig, and Gian Paolo Vacca.  Critical reflections on asymptotically safe gravity.  Front. in Phys., 8:269, 2020, 2004.06810.
  * [9] J. Ambjørn, A. Goerlich, J. Jurkiewicz, and R. Loll.  Nonperturbative Quantum Gravity.  Phys.Rept., 519:127–210, 2012, 1203.3591.
  * [10] R. Loll.  Quantum Gravity from Causal Dynamical Triangulations: A Review.  Class. Quant. Grav., 37(1):013002, 2020, 1905.08669.
  * [11] K.G. Wilson and John B. Kogut.  The Renormalization group and the epsilon expansion.  Phys.Rept., 12:75–200, 1974.
  * [12] Franz J. Wegner and Anthony Houghton.  Renormalization group equation for critical phenomena.  Phys. Rev., A8:401–412, 1973.
  * [13] Tim R. Morris and Anthony W. H. Preston.  Manifestly diffeomorphism invariant classical Exact Renormalization Group.  JHEP, 06:012, 2016, 1602.08993.
  * [14] Tim R. Morris.  A Manifestly gauge invariant exact renormalization group.  In The exact renormalization group. Proceedings, Workshop, Faro, Portugal, September 10-12, 1998, pages 1–40, 1998, hep-th/9810104.
  * [15] Tim R. Morris.  A Gauge invariant exact renormalization group. 1.  Nucl. Phys., B573:97–126, 2000, hep-th/9910058.
  * [16] Tim R. Morris.  A Gauge invariant exact renormalization group. 2.  JHEP, 12:012, 2000, hep-th/0006064.
  * [17] Tim R. Morris.  An Exact RG formulation of quantum gauge theory.  Int. J. Mod. Phys., A16:1899–1912, 2001, hep-th/0102120.
  * [18] S. Arnone, Yu. A. Kubyshin, T. R. Morris, and J. F. Tighe.  Gauge invariant regularization in the ERG approach.  In High energy physics and quantum field theory. Proceedings, 15th International Workshop, QFTHEP 2000, Tver, Russia, September 14-20, 2000, pages 297–304, 2000, hep-th/0102011.
  * [19] S. Arnone, Yu. A. Kubyshin, T. R. Morris, and J. F. Tighe.  A Gauge invariant regulator for the ERG.  Int. J. Mod. Phys., A16:1989, 2001, hep-th/0102054.
  * [20] Stefano Arnone, Yuri A. Kubyshin, Tim R. Morris, and John F. Tighe.  Gauge invariant regularization via SU($N|N$).  Int. J. Mod. Phys., A17:2283–2330, 2002, hep-th/0106258.
  * [21] Stefano Arnone, Antonio Gatti, and Tim R. Morris.  Towards a manifestly gauge invariant and universal calculus for Yang-Mills theory.  Acta Phys. Slov., 52:621–634, 2002, hep-th/0209130.
  * [22] Stefano Arnone, Antonio Gatti, and Tim R. Morris.  A Manifestly gauge invariant exact renormalization group.  In Renormalization group. Proceedings, 5th International Conference, RG 2002, Tatranska Strba, Slovakia, March 10-16, 2002, 2002, hep-th/0207153.  [Submitted to: Acta Phys. Slov.(2002)].
  * [23] Stefano Arnone, Antonio Gatti, and Tim R. Morris.  A Proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory.  Phys. Rev., D67:085003, 2003, hep-th/0209162.
  * [24] Stefano Arnone, Tim R. Morris, and Oliver J. Rosten.  A Generalised manifestly gauge invariant exact renormalisation group for SU(N) Yang-Mills.  Eur. Phys. J., C50:467–504, 2007, hep-th/0507154.
  * [25] Stefano Arnone, Antonio Gatti, and Tim R. Morris.  Manifestly gauge invariant computations.  In Renormalization group. Proceedings, 5th International Conference, RG 2002, Tatranska Strba, Slovakia, March 10-16, 2002, 2002, hep-th/0207154.  [Submitted to: Acta Phys. Slov.(2002)].
  * [26] Antonio Gatti.  A Gauge invariant flow equation.  PhD thesis, Southampton U., 2002, hep-th/0301201.
  * [27] Tim R. Morris and Oliver J. Rosten.  A Manifestly gauge invariant, continuum calculation of the SU(N) Yang-Mills two-loop beta function.  Phys. Rev., D73:065003, 2006, hep-th/0508026.
  * [28] Oliver J. Rosten, Tim R. Morris, and Stefano Arnone.  The Gauge invariant ERG.  In 13th International Seminar on High-Energy Physics: Quarks 2004 Pushkinskie Gory, Russia, May 24-30, 2004, 2004, hep-th/0409042.
  * [29] Oliver Jacob Rosten.  The Manifestly gauge invariant exact renormalisation group.  PhD thesis, Southampton U., 2005, hep-th/0506162.
  * [30] Oliver J. Rosten.  A Primer for manifestly gauge invariant computations in SU(N) Yang-Mills.  J. Phys., A39:8699–8726, 2006, hep-th/0507166.
  * [31] Oliver J. Rosten.  A Manifestly gauge invariant and universal calculus for SU(N) Yang-Mills.  Int. J. Mod. Phys., A21:4627–4762, 2006, hep-th/0602229.
  * [32] Oliver J. Rosten.  General Computations Without Fixing the Gauge.  Phys. Rev., D74:125006, 2006, hep-th/0604183.
  * [33] Stefano Arnone, Tim R. Morris, and Oliver J. Rosten.  Manifestly Gauge Invariant Exact Renormalization Group.  Fields Inst. Commun., 50:1, 2007, hep-th/0606181.
  * [34] Oliver J. Rosten.  Fundamentals of the Exact Renormalization Group.  Phys. Rept., 511:177–272, 2012, 1003.1366.
  * [35] Oliver J. Rosten.  Aspects of Manifest Gauge Invariance.  PoS, FACESQCD:035, 2010, 1102.3091.
  * [36] Kevin Falls and Tim R. Morris.  Conformal anomaly from gauge fields without gauge fixing.  Phys. Rev., D97(6):065013, 2018, 1712.05011.
  * [37] Matthew Kellett and Tim R Morris.  Parisi-sourlas supergravity.  Classical and Quantum Gravity, 37(19):195018, sep 2020.
  * [38] Kevin Falls.  Background independent exact renormalisation.  Eur. Phys. J. C, 81(2):121, 2021, 2004.11409.
  * [39] A. A Slavnov.  The Pauli-Villars Regularization for Nonabelian Gauge Theories.  Teor. Mat. Fiz., 33:210–217, 1977.
  * [40] Timur D. Bakeyev and A. A. Slavnov.  Higher covariant derivative regularization revisited.  Mod. Phys. Lett., A11:1539–1554, 1996, hep-th/9601092.
  * [41] Manuel Asorey and Fernando Falceto.  Geometric regularization of gauge theories.  Nuclear Physics B, 327(2):427–460, nov 1989.
  * [42] Emil Mottola.  Functional integration over geometries.  J. Math. Phys., 36:2470–2511, 1995, hep-th/9502109.
  * [43] B.S. DeWitt and D.W. Bryce Seligman.  The Global Approach to Quantum Field Theory.  Number v. 1 in International series of monographs on physics. Oxford University Press, 2003.
  * [44] L.F. Abbott.  Introduction to the Background Field Method.  Acta Phys.Polon., B13:33, 1982.
  * [45] F. J. Wegner.  Some invariance properties of the renormalization group.  J. Phys., C7:2098, 1974.
  * [46] Jose I. Latorre and Tim R. Morris.  Exact scheme independence.  JHEP, 11:004, 2000, hep-th/0008123.
  * [47] Jose I. Latorre and Tim R. Morris.  Scheme independence as an inherent redundancy in quantum field theory.  Int. J. Mod. Phys., A16:2071–2074, 2001, hep-th/0102037.
  * [48] L. D. Faddeev and A. A. Slavnov.  Gauge Fields. Introduction to Quantum Theory., volume 50.  1980\.
  * [49] Manuel Asorey and Fernando Falceto.  On the consistency of the regularization of gauge theories by high covariant derivatives.  Phys. Rev., D54:5290–5301, 1996, hep-th/9502025.
  * [50] Stefano Arnone, Antonio Gatti, and Tim R. Morris.  Exact scheme independence at one loop.  JHEP, 05:059, 2002, hep-th/0201237.
  * [51] Stefano Arnone, Antonio Gatti, Tim R. Morris, and Oliver J. Rosten.  Exact scheme independence at two loops.  Phys. Rev., D69:065009, 2004, hep-th/0309242.
  * [52] Oliver J. Rosten.  Scheme independence to all loops.  J. Phys., A39:8141–8156, 2006, hep-th/0511107.
  * [53] Joseph Polchinski.  Renormalization and Effective Lagrangians.  Nucl.Phys., B231:269–295, 1984.
  * [54] Brian J. Warr.  Renormalization of Gauge Theories Using Effective Lagrangians. 1.  Annals Phys., 183:1, 1988.