11institutetext: Lab AIM, CEA/CNRS/Université Paris-Saclay, Université de
Paris, F-91191 Gif-sur-Yvette, France 22institutetext: Laboratoire de Physique
Nucléaire et des hautes énergies, Sorbonne Université, 4 Place Jussieu, 75005
Paris, France
22email<EMAIL_ADDRESS>33institutetext: European Southern
Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München, Germany
33email<EMAIL_ADDRESS>44institutetext: Center for Astrophysics and
Space Sciences, University of California, San Diego, 9500 Gilman Drive, La
Jolla, CA 92093, USA 44email<EMAIL_ADDRESS>55institutetext: Institut für
Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen,
Germany 66institutetext: Dr. Karl Remeis-Sternwarte and Erlangen Centre for
Astroparticle Physics, Friedrich-Alexander Universität Erlangen-Nürnberg,
Sternwartstr. 7, 96049 Bamberg, Germany 77institutetext: CRESST, CSST,
Department of Physics, University of Maryland Baltimore County, 1000 Hilltop
Circle, Baltimore, MD 21250, USA 88institutetext: NASA Goddard Space Flight
Center, Astrophysics Science Division, Code 661, Greenbelt, MD 20771, USA

# Potential origin of the state-dependent high-energy tail in the black hole
microquasar Cygnus X-1 as seen with INTEGRAL

F. Cangemi 1122 T. Beuchert 33 T. Siegert 44 J. Rodriguez 11 V. Grinberg 55 R.
Belmont 11 C. Gouiffès 11 I. Kreykenbohm 66 P. Laurent 11 K. Pottschmidt 7788
J. Wilms, 66

(Accepted for publication)

###### Abstract

Context.  0.1–10 MeV observations of the black hole microquasar Cygnus X-1
have shown the presence of a spectral feature in the form of a power law in
addition to the standard black body (0.1–10 keV) and Comptonization (10–200
keV) components observed by INTEGRAL in several black-hole X-ray binaries.
This so-called “high-energy tail” was recently shown to be strong in the hard
spectral state of Cygnus X-1, and, in this system, has been interpreted as the
high-energy part of the emission from a compact jet.

Aims. This result was nevertheless obtained from a data set largely dominated
by hard state observations. In the soft state, only upper limits on the
presence and hence the potential parameters of a high-energy tail could be
derived. Using an extended data set, we aim to obtain better constraints on
the properties of this spectral component in both states.

Methods. We make use of data obtained from about 15 years of observations with
the INTEGRAL satellite. The data set is separated into the different states
and we analyze stacked state-resolved spectra obtained from the X-ray monitors
(JEM-X), the gamma-ray imager (IBIS), and the gamma-ray spectrometer (SPI)
onboard.

Results. A high-energy component is detected in both states, confirming its
earlier detection in the hard state and its suspected presence in the soft
state with INTEGRAL, as seen in a much smaller SPI data set. We first
characterize the high-energy tail components in the two states through a
model-independent, phenomenological analysis. We then apply physical models
based on hybrid Comptonization (eqpair and belm). The spectra are well modeled
in all cases, with a similar goodness of the fits. While in the semi-
phenomenological approach the high-energy tail has similar indices in both
states, the fits with the physical models seem to indicate slightly different
properties. Based on this approach, we discuss the potential origins of the
high-energy components in both the soft and hard states, and favor an
interpretation where the part of the high-energy component is due to a compact
jet in the hard state and hybrid Comptonization in either a magnetized or
nonmagnetized corona in the soft state.

###### Key Words.:

Accretion, accretion disks — Physical data and processes — Black hole physics
— X-rays: binaries — Stars: black holes — Stars: individual (Cyg X-1)

††offprints<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Black-hole X-ray binaries (BHXBs) are systems with variable X-ray emission and
can transit through various spectral states during outbursts. Because of the
discovery of different types of radio relativistic jets in these systems
(e.g., Mirabel et al., 1992; Fender et al., 1999), they are commonly dubbed
“microquasars”. The two main (canonical) states are the low-hard state (LHS)
and the high-soft state (HSS; see, e.g., Remillard & McClintock, 2006;
Belloni, 2010, for a precise definition of the states). In the LHS, the X-ray
spectrum between ${\sim}1$ and ${\sim}100$ keV is dominated by a hard power
law with a photon index of $\Gamma\lesssim 1.5$ which cuts off at a few
hundred keV. Compact radio jets are only observed in the LHS (e.g., Fender,
2001) and have been resolved in a couple of sources: Cygnus X-1 (Cyg X-1) and
GRS 1915+105 (e.g., Stirling et al., 2001; Fuchs et al., 2003). In the HSS,
the spectrum is dominated by a blackbody component peaking at ${\sim}1\,$keV.
This component is associated with thermal emission from an optically thick
accretion disk. Usually, no radio emission is detected in this state (e.g.,
Fender et al., 1999; Remillard & McClintock, 2006; Belloni, 2010), with the
possible exception of Cyg X-1 where tenuous radio emission was detected using
very long baseline interferometry (Rushton et al., 2012).

In several bright sources, observations made at higher energies reveal the
presence of a high-energy tail above a few hundred keV that can sometimes
extend to several MeV (Grove et al., 1998). In this paper we seek further
insight into the state-resolved high-energy tail emission in order to discuss
its physical origin which is still subject to much debate. To do so, we focus
on the most promising candidate, Cyg X-1, and exploit the combined
capabilities of the Joint European X-Ray Monitor (JEM-X, Lund et al., 2003),
the Imager on Board the INTEGRAL Satellite (IBIS; Lebrun et al., 2003), and
the SPectrometer onboard INTEGRAL (SPI; Vedrenne et al., 2003) to perform
detailed spectral analysis of the broad-band $\sim$3 keV–2 MeV data by
accumulating all existing INTEGRAL data of this source taken over the first 15
years of the satellite’s life.

Although it was used as a prototype for the definition of the (canonical)
X-ray spectral states, Cyg X-1 is a peculiar case. Located at a distance of
$1.86\pm 0.12$ kpc (Reid et al., 2011), it is a persistent high-mass X-ray
binary (HMXB) with a $14.8\pm 1.0\,M_{\odot}$ black hole orbiting a blue
supergiant star with a mass of $19.2\pm 1.9\,M_{\odot}$ (Orosz et al., 2011).
The orbital plane is seen at an inclination of ${\sim}27$° (Orosz et al. 2011,
Miller-Jones et al., submitted), a value that we use throughout this paper. In
contrast to the majority of (transient) BHXBs, its individual states persist
on much longer timescales (e.g., Grinberg et al., 2013), and even though the
parameters ofx its spectral states are similar to those of other sources, Cyg
X-1 does not follow the typical “q-track” pattern for transients: for example,
it has never returned to quiescence.

A high-energy tail above the thermal Comptonization continuum in Cyg X-1 was
detected for the first time in both the soft and hard accretion states by the
COMPTEL instrument (McConnell et al., 2000, 2002) on board the Compton Gamma-
Ray Observatory (CGRO). The tail in the LHS was later confirmed with the SPI
(Cadolle Bel et al., 2006; Jourdain et al., 2012) and IBIS (Laurent et al.,
2011; Rodriguez et al., 2015, hereafter L11, R15), but so far has remained
undetected by INTEGRAL in the HSS (Jourdain et al., 2014, hereafter J14). This
might be both due to the intrinsic faintness of the component and the rather
short exposure time of INTEGRAL in the HSS. Observations obtained between 2003
and 2011 contained $\sim 90$ % of LHS data (e.g., R15). Thanks to a spectral
transition to a long-lasting HSS in 2011 (Grinberg et al., 2011), follow-up
studies such as the present one will be increasingly sensitive to the high-
energy tail emission of Cyg X-1 in the HSS.

Detection and investigation of the high-energy tails in both states of Cyg X-1
with INTEGRAL are possible thanks the mission’s combination of a large
effective area with the capability to measure polarization – a key tool for
breaking degeneracies in the quest for the physical origin of the high-energy
tail emission. Several studies have confirmed a high degree of polarization
for the high-energy tail in the LHS (L11, Jourdain et al., 2012, and R15 for a
state-resolved polarimetric analysis), which is compatible with synchrotron
emission from a jet (observed in the source’s LHS e.g., Stirling et al., 2001;
Zdziarski et al., 2012). However, this extension above the Comptonized
spectrum could also originate from a nonthermal electron distribution in the
corona (e.g., McConnell et al., 2002; Romero et al., 2014; Jourdain et al.,
2014; Chauvin et al., 2018). In particular, Del Santo et al. (2013) separated
all Cyg X-1 INTEGRAL data obtained until 2009 into 12 separate sub-states
according to their hardness in the IBIS range and their 30–100 keV spectral
slope. They studied these 12 stacked spectra within the framework of hybrid
models, either nonmagnetized (eqpair) or with magnetization (belm), and were
able to obtain upper limits on the corona’s magnetic field.

We are therefore not only interested in assessing the existence of a high-
energy tail in the HSS but also in investigating its origin with the extended
amount of data available. Our approach, accumulating data until 2017,
increases the count statistics by a significant amount over all earlier works
and allows us to study the parameter space of hybrid models not only in the
LHS but also, for the first time, with INTEGRAL in the HSS up to 2 MeV. This
paper is organized as follows. In Sect. 2, we outline the selection criteria
for data considered in this work and the extraction of data products. In Sect.
3, we present a set of techniques and models used to describe the data
products. In Sect. 4, we discuss the results and in Sect. 5 we provide our
conclusions.

##  2 Observations and data reduction

###  2.1 State classification and data selection

We consider all Cyg X-1 observations made with INTEGRAL from 2003 June 9 to
2017 November 7 (Modified Julian Day, MJD 52799 to 58054) corresponding to the
satellite revolutions (hereafter revs) 54 to 1882. We restrict our analysis to
“science windows” (hereafter scws), that is, individual INTEGRAL pointings of
typically 1800–3600 s duration, where Cyg X-1 is less than 10° off axis. This
results in a total of 7288 scws corresponding to 11.4 Ms of dead time-
corrected exposure.

Similar to R15, we separate the whole data set into three states (soft/HSS,
intermediate/IS, and hard/LHS) using the model-independent classification of
Grinberg et al. (2013). This classification defines LHS, HSS, and IS based on
count rates and hardnesses measured by all-sky monitor instruments (RXTE/ASM,
MAXI, and Swift/BAT). Figure 1 shows the classification of the individual
scws. This figure extends the similar Fig. 1 from Grinberg et al. (2013), to
whom we refer for the details of the methodology of the classification. As in
R15, we use all-sky monitor data within 6 hours from a given INTEGRAL scw for
the classification. Such stringent time constraints on the classification are
necessary because the source is known to undergo fast state transitions (Böck
et al., 2011; Grinberg et al., 2013). In total, we roughly double the total
exposure time in the LHS compared to R15, and the new extended HSS covering
roughly MJD 55800 to 57400 (with some excursions into the LHS, Fig. 1) allows
us to increase the HSS exposure by a factor of three compared to R15. After
MJD 57400, INTEGRAL mostly catches Cyg X-1 in the LHS (Fig. 1). To summarize,
we obtain 3178, 2262, and 600 scws, in the LHS, HSS, and IS respectively,
corresponding to 4.4 Ms, 3.4 Ms, and 0.9 Ms of exposure time. A total of 1248
scws remain unclassified with this method and are not taken into account in
our analysis.

Figure 1: Spectral state evolution of Cyg X-1 between 2003 and 2017 after
Grinberg et al. (2013). The HSS is shown in red, the IS in green, and the LHS
in blue. Unclassified data are plotted in gray. Upper panel: 30–50 keV
INTEGRAL/ISGRI light curve. The vertical dashed line indicates the change of
the INTEGRAL data analysis software. Lower three panels: RXTE/ASM, MAXI, and
Swift/BAT light curves of Cyg X-1.

###  2.2 Spectral extraction: INTEGRAL/IBIS/ISGRI

Data from INTEGRAL/IBIS’ upper detector layer ISGRI111The INTEGRAL’s Soft
Gamma-Ray Imager (Lebrun et al., 2003) taken before December 2015 (INTEGRAL
revolution 1627, MJD 57387) were reduced with version 10.2 of the Off-line
Scientific Analysis (OSA) software. Data obtained after that date were
extracted with OSA v11. This split is necessary because (at the time of
writing) specific versions of calibration data —specifically for the gain
correction of the instrumental energy scale— are required for both software
versions222See https://www.isdc.unige.ch/integral/analysis#Software for the
restrictions on the use of OSA v11 with IBIS.. The long-term IBIS light curve
(with a resolution of individual scws) is shown in the top panel of Fig. 1.
The different gain correction used before and after December 2015 is seen as
an overall increase in the 30–50 keV count rate. For comparison, the 30–50 keV
Crab count rate is ${\sim}87\,\mathrm{counts}\,\mathrm{s}^{-1}$ for OSA v10.2
and ${\sim}134\,\mathrm{counts}\,\mathrm{s}^{-1}$ for OSA v11.

IBIS/ISGRI data were reduced with the standard
procedure333https://www.isdc.unige.ch/integral/download/osa/doc/11.1/osa_um_ibis/man.html;.
For each satellite revolution we first produce a mosaic image to identify the
brightest sources (here defined as those with a detection significance greater
than 7$\sigma$ in the 30–50 keV mosaic image) of the field. We then extract
light curves and spectra on a scw-by-scw basis. For each scw, we construct a
sky model including the brightest sources as defined above (usually Cyg X- 3,
Cyg X-2, GRO J2058+42, and KS 1947+300, in addition to Cyg X-1). Taking the
background into account, the analysis software then reconstructs the sky image
and corresponding source count rates by deconvolving the shadowgrams projected
on the detector plane. The roughly 250 scws from the time period of the 2015
June outburst of V404 Cyg (INTEGRAL revolutions 1554 through 1558) remain
unclassified and were therefore ignored in our analysis.

Source spectra were extracted on a logarithmic grid of 60 spectral channels.
In OSA v11.0, the spectral binning and the response matrices are calculated
and provided automatically by the extraction software. In OSA v10.2, the
binning is provided by the user and based on a pre-rebinned redistribution
matrix file (rmf). We construct the latter to be as close as possible to the
OSA v11.0 one, resulting in spectral bin deviations of less than 0.25 keV per
channel. We then stack the individual spectra according to the version of the
extraction software and following their spectral state classification. This
results in a total of four stacked spectra: two per spectral state. In all
spectra, we add 2 % systematic uncertainty to each of the channels3. For the
spectral fittings, we omit the spectral channels below 25 keV for all the
ISGRI spectra obtained with OSA v10.2 and those below 30 keV for the OSA 11.0
ones.

###  2.3 Spectral extraction: INTEGRAL/SPI

The SPI data analysis relies on the comparison of sky models and a description
of the instrumental background to data in the raw format of counts per energy
bin, detector, and pointing (scw). See Vedrenne et al. (2003) for details
about the instruments. For the SPI extraction we include the source-intrinsic
and background variability in the sky model, yielding state-resolved, long-
term spectra. In the case of Cyg X-1 our sky models consist of point-sources
in the $16^{\circ}\times 16^{\circ}$ field of view with time-dependent fluxes.
Above ${\sim}100$ keV, only Cyg X-1 and Cyg X-3 are significantly detected and
taken into account.

First, we collect the good time intervals as defined for the IBIS/ISGRI data
analysis divided into soft and hard state. Because SPI is very sensitive to
instrument mode changes, (solar) particle events, detector failures, and Van
Allen belt passages, we apply additional filter criteria as shown in Table 1.
The resulting number of pointings is 2358 and 1771 for the LHS and HSS,
respectively, with a total dead-time-corrected observation time of 3.4 Ms and
4.7 Ms, respectively. The resulting exposure map using all good pointings is
shown in Fig. 2. This map shows Cyg X-1 in the center of the chosen exposure
and which (known) sources are exposed and how they might contribute in the
fit.

Table 1: Selection criteria for the SPI data set based on “science housekeeping” (HK) parameters. Revolutions 1554 to 1558 are excluded because of the V404 Cygni outburst. HK parameter | Allowed values/ranges
---|---
SPIMode | 41
ACSRate/GeSatTot | $\left[360,400\right]$
HVDetMean | $\left[2.05,4.05\right]$
OrbitRevPhase | $\left[0.1,0.9\right]$
$\Delta$TempColdPlt | $\left[0.5,0.8\right]$
FracGoodEvtsTot | $\left[0.04,0.06\right]$
IREM TC2Rate | $<3$
IREM S32Rate | $<5$
Revolution | $\lnot\left\\{236,354,1147,1149,1604\right\\}$
| $\lnot\left\\{1554,\ldots,1558\right\\}$

We consider single events (SEs) below 530 keV and pulse-shape discriminated
(PSD) events above that energy. This removes strong “electronic noise”
features between 1.4 and 1.7 MeV and allows for a smooth spectral
extraction444https://www.isdc.unige.ch/integral/download/osa/doc/11.1/osa_um_spi/man.html.
We also extract SE spectra up to 755 keV to establish consistency between the
two data types. The derived PSD fluxes are then scaled by a factor of 1/0.85
to account for the change in efficiency 4.

Figure 2: SPI exposure of Cyg X-1 in its soft state in units of
$10^{6}\,\mathrm{cm^{2}\,s^{-1}}$. The fully coded field of view of SPI is
$16^{\circ}\times 16^{\circ}$. Green points indicate point sources that have
to be taken into account in the SPI data analysis. Red points show other
sources listed in the SPI catalog of Bouchet et al. (2011) but only seen below
$\lesssim$100 keV or at specific flare times.

For the spectral extraction, we define 21 logarithmic energy bins between 25
keV and 755 keV for the SE range, and 8 logarithmic bins between 530 keV and
2000 keV for the PSD range. The spectra are obtained bin by bin from maximum-
likelihood fits accounting for Poisson-distributed data. These fits include
the two-component background model as described by Siegert et al. (2019),
which is based on the SPI BackGround and Response DataBase (BGRDB, Diehl et
al., 2018), as well as Cyg X-1 and Cyg X-3. We use the OSA spimodfit (Strong
et al., 2005) to perform these fits. Given the large unknown in the SPI data
reduction processes, and although Cyg X-1 is not formally a source
sufficiently bright to present the caveats described by Roques & Jourdain
(2019) in both states, we also extract SPI spectra using different methods
(see Appendix B).

The SPI background variation timescale depends on energy and energy bin size
(Siegert et al., 2019); additionally, the absolute fluxes of the sources may
change with time, even when they are in the same state. To determine the
optimal variation timescales for both background and source at the same time,
we therefore perform a grid search. To do so, we define 13 variability
timescales, between one pointing (${\sim}$0.6 h) and no variability (constant
background and source), and double the times in each step (in total 169 grid
points). The smaller the variability timescale, the more parameters are
included in the fit. Although this results in a better likelihood, it tends to
over-sample the data. In order to penalize large numbers of parameters, we
search for the minimum Akaike information criterion (AIC Akaike, 1974;
Cavanaugh et al., 1997) instead, which defines the fit quality of models
relative to each other (see the Appendix for examples of AIC maps). From this
analysis, we find that in the HSS the background variability timescale is
rather constant as a function of energy, changing between 12 h and 72 h
(${\sim}1/6$–1 INTEGRAL orbit). In the LHS, background variations are more
frequent. The background variability timescale is found to decrease down to
the pointing timescale below 150-200 keV. However, at these energies the
background and source patterns in addition to the large number of fitted
parameters are too degenerate to determine a robust estimate for a temporal
variability timescale of Cyg X-1.

It appears that switching off a strong source variability provides similar
flux normalizations to IBIS/ISGRI spectral extractions
($c_{\mathrm{ISGRI}}/c_{\mathrm{SPI}}\approx 1.0$, see Table 2) in the LHS,
but to a lesser extent in the HSS ($c_{\mathrm{ISGRI}}/c_{\mathrm{SPI}}\approx
1.3$). Above 300 keV (400 keV) in soft (hard) state, the number of fitted
parameters for background and source are considerably smaller, also reducing
the degeneracy in the maximum likelihood fits. The source spectra are then
found to be steady, and the high-energy tail (Sect. 3.2) is constant in time.

We compare our extracted INTEGRAL/SPI hard state spectrum with the
INTEGRAL/IBIS Compton-mode spectrum published by R15 in Fig. 3. At the time,
the spectrum was blamed for being very hard. Here, we observe a softer
spectrum as extracted with SPI compared to the Compton-mode spectrum
previously published by R15.

Figure 3: Comparison of our extracted hard-state INTEGRAL/SPI spectrum with
INTEGRAL/IBIS/PICsIT spectra (a) extracted for the same time period as SPI and
(b) for data before 2012 as published by Rodriguez et al. (2015).

###  2.4 Spectral extraction: INTEGRAL/JEM-X

For the Joint European X-ray Monitor spectral extraction, we restrict our
selection to scws where the source is in the field of view of the JEM-X unit
1, which is where the source is less than 5∘ off-axis. This selection results
in 836 and 1112 scws for the LHS and HSS, respectively. The data from JEM-X 1
are reduced with OSA v11.0. We follow the standard steps described in the
JEM-X user
manual555https://www.isdc.unige.ch/integral/download/osa/doc/11.1/osa_um_jemx/man.html..
Spectra are extracted in each scw where the source is automatically detected
by the software at the image creation and fitting step, and are then computed
from $\sim$3 to $\sim$34 keV with 32 logarithmic bins. The individual spectra
are then combined with the OSA spe_pick tool. The appropriate ancillary
response files (arfs) are produced during the spectral extraction and combined
with spe_pick, while the redistribution matrix file (rmf) is rebinned from the
instrument characteristic standard rmf with j_rebin_rmf. We add 3 % systematic
error on all spectral channels for each of the stacked spectra, as recommended
in the JEM-X user manual, and the spectra are considered between 3 and 25 keV.

##  3 Results

###  3.1 Methodology for the spectral analysis

We use both the Interactive Spectral Interpretation System (ISIS; Houck &
Denicola, 2000) and XSPEC V.12.10 (Arnaud, 1996) packages for the spectral
analysis.

In order to assess the presence, significance, and main properties of the
high-energy tail, we first analyze the data with a simple phenomenological
model, that is, a thermal Comptonization continuum in combination with a power
law extending into the MeV range (Sect. 3.2). Keeping in mind the suggested
origin for the LHS high-energy tail proposed by L11, Jourdain et al. (2012),
and R15, we then investigate other possible origins for this component. We use
physical models considering the broad-band source spectrum as originating from
a single medium composed of a hybrid thermal/nonthermal particle distribution
(Sect. 3.3 and 3.4).

Figure 4: Long-term (2003–2018), broad-band (3 keV to 2 MeV), and state-
resolved (red: HSS, blue: LHS) stacked spectra of JEM-X 1 (light colors),
ISGRI (intermediate colors), and SPI (darker colors).

###  3.2 Phenomenological approach: a high-energy tail in both states

The LHS and HSS stacked spectra of JEM-X 1, IBIS/ISGRI, and SPI are shown in
Fig. 4. Following the approach presented in J14 and R15, we first focus on the
3-400 keV spectral band. In that energy range, we apply the thermal
Comptonization model compTT (Titarchuk, 1994) being reflected off the disk,
which in xspec notation is

$\texttt{const*reflect(compTT)},$

and has been successfully used by J14 and R15. The const component is a
multiplicative constant necessitated by the differences in absolute flux
calibration between the different instruments used. In the LHS, we freeze the
value of the seed photon temperature of the comptt component to $kT_{0}=0.2$
keV (e.g., Balucinska & Hasinger, 1991; Wilms et al., 2006, R15). In the HSS,
we also need to add the thermal emission from the disk (diskbb) and an iron
line at 6.4 keV (fitted by a gaussian, gauss). Thus, the model is written in
xspec as

$\texttt{const*(reflect(compTT)+diskbb+gauss)}.$

As the spectral resolution of JEM-X prevents a fine description of the iron
line, we fix its energy and its width to $E_{\mathrm{Fe}}=6.4$ keV and
$\sigma_{\mathrm{Fe}}=0.4$ keV, respectively. The seed photon temperature of
the comptt component is fixed to the temperature of the diskbb component.

Once an appropriate fit has been obtained with the spectra below 400 keV, we
then add ISGRI and SPI data above 400 keV and let the parameters vary freely.
A clear high-energy excess is visible in both states above the continuum
described with the above-mentioned models. We model this additional emission
with a power-law component. We assess the significance of the added power-law
tail using an F-test. A low P-value probability of the F-test suggests that
the null-hypothesis, that is, the assumption that the thermal Comptonization
continuum describes the broad-band spectrum well, is false and means that the
fit requires the additional power-law component to model the high-energy tail.
We find an F-test probability of $P=3.5\times 10^{-24}$ and $2.6\times
10^{-4}$ in the LHS and HSS, respectively. Table 2 lists all best-fit
parameters in the 3–2000 keV range and Fig. 5 shows the best-fit models and
their residuals for both states.

Table 2: Model parameters obtained with the phenomenological fits to the LHS and HSS 3–2000 keV stacked spectra. Fixed parameters are denoted with an asterisk.  Parameters | LHS | HSS
---|---|---
$c_{\mathrm{JEM-X}}$ | 1∗ | 1∗
$c_{\mathrm{ISGRI,\texttt{OSA10}}}$ | $0.85^{+0.04}_{-0.04}$ | –
$c_{\mathrm{ISGRI,\texttt{OSA11}}}$ | $0.88^{+0.04}_{-0.04}$ | $0.89^{+0.03}_{-0.03}$
$c_{\mathrm{SPI}}$ | $0.90^{+0.04}_{-0.04}$ | $1.20^{+0.00}_{-0.02}$
$kT_{0}$ [keV] | 0.2$\ast$ | $0.9^{+0.1}_{-0.1}$
$kT$ [keV] | $53^{+3}_{-3}$ | $62^{+71}_{-28}$
$\tau$ | $1.47^{+0.09}_{-0.09}$ | $0.29^{+0.02}_{-0.02}$
$\Gamma_{\mathrm{tail}}$ | $2.06^{+0.03}_{-0.03}$ | $2.0^{+0.3}_{-0.4}$
$\Omega/2\pi$ | $0.8^{+0.2}_{-0.1}$ | $0.5^{+0.2}_{-0.2}$
$E_{\mathrm{Fe}}$ [keV] | – | 6.4∗
$\sigma_{\mathrm{Fe}}$ [keV] | – | 0.4∗
F${}_{0.4-2\mathrm{\,MeV}}$ | – | –
[$\times 10^{-9}$ ergs. cm-2. s-1] | 3.8 | 0.9
$\chi^{2}_{\mathrm{red}}$ | 141.54/117 | 103.29/78
Figure 5: Best-fit models obtained from our phenomenological fitting of the
3–2000 keV LHS (blue) and HSS (red) spectra. JEM-X, ISGRI, and SPI data are
represented with light, intermediate, and darker colors respectively. The
different model components are shown with different line styles: Comptonized
continuum (densely dashed), additional power law (dotted), disk emission
(dotted-dashed), and the iron line (loosely dashed). The bottom panels show
the residuals between these models and spectra for the source in both states.

####  3.2.1 The LHS

In the LHS, the electron temperature, $kT\sim 53$ keV, is consistent with the
value obtained by J14 and roughly consistent with that of R15. The optical
depth, $\tau\sim 1.5$, seems here slightly higher than in the two previous
studies. The reflected photon fraction of $\Omega/2\pi\sim 0.8$ is consistent
with the one found by J14 but remains higher than in R15 (where
$\Omega/2\pi=0.13\pm 0.02$). These small differences are probably due to the
intrinsically variable nature of the source, our largely extended observations
sample and integrated exposure time in this state compared to these previous
studies, and the extended spectral range we consider here compared to J14 and
R15, and are therefore not surprising. As expected from the comparison of our
SPI spectrum with that of R15 (Fig. 3), we find a softer photon index
($\Gamma_{\mathrm{tail}}\sim$ 2.06; Table 2) for the additional power-law
component than the (very hard) value found by R15
($\Gamma_{\mathrm{tail}}^{\mathrm{R15}}$=$1.4^{+0.2}_{-0.3}$). Our results
nevertheless remain rather consistent with those of J14
($\Gamma_{\mathrm{tail}}^{\mathrm{J14}}$=$1.8\pm 0.2$).

####  3.2.2 The hard-soft state

Significant residuals arise in the HSS between tens to hundreds of keV for
IBIS/ISGRI when extracted with OSA v10.2. We therefore exclude the OSA v10.2
HSS data from the fit. The resulting reduction in count statistics has only a
minor impact due to the coverage of the same energy range by SPI.

We are able to integrate a sufficient number of counts in the HSS of Cyg X-1
and can confirm the suggestion previously made by J14 of a possible high-
energy tail in the HSS. We find a photon index for the additional power law
$\Gamma_{\mathrm{tail}}\simeq 2.0$ (Table 2), which is consistent with the
value we find in the LHS. We find an electron temperature of $62^{+71}_{-28}$
, which is not well constrained, and an optical depth of $\tau\simeq 0.29$
consistent with the previous results of J14. We can constrain the reflection
component to $\Omega/2\pi\simeq 0.5$ (Table 2) .

###  3.3 State-resolved spectral modeling: unmagnetized hybrid plasma

####  3.3.1 Basics of the eqpair model

To go beyond the simple phenomenological approach, we model the data with the
emission from a single region, where the electron distribution is hybrid. In
such a model, the high-energy tail is explained by the emission of a
nonthermal population of accelerated particles. These particles cool down and
eventually thermalize because of several processes such as Coulomb collisions.
The steady-state distribution is then a hybrid distribution constituted by a
low-energy, thermal population producing the bulk of the emission below 400
keV and a high-energy, nonthermal population producing the high-energy tail
above 400 keV.

To model such a hybrid plasma, we first use eqpair (Coppi, 1999), which has
been successfully used in describing the broad-band spectra of Cyg X-1 in
different states (Gierliński et al., 1999; McConnell et al., 2002; Wilms et
al., 2006; Cadolle Bel et al., 2006; Fritz et al., 2007; Nowak et al., 2011;
Del Santo et al., 2013). This model takes into account all microphysical
interactions and processes between particles and photons (bremsstrahlung,
Compton-scattering, pair production, and electron-electron Coulomb collisions)
in a homogeneous, isotropic, and ionized medium of total optical depth
$\tau_{\mathrm{T}}$ which is the sum of the proton Thomson optical depth
$\tau_{\mathrm{p}}$ plus the optical depth of electron-positron pairs. Soft
photons are injected into the medium as a multicolor disk black-body
characterized by the inner disk temperature $\mathrm{k}T_{\mathrm{bb}}$ and
the total power $L_{s}$. Energy is provided to the electron population through
two channels: either directly in the thermal pool of electrons with power
$L_{\mathrm{th}}$, or as nonthermal particles with a power-law distribution
with index $\Gamma_{\mathrm{inj}}$ and power $L_{\mathrm{nth}}$. The total
power injected into electrons is
$L_{\mathrm{h}}=L_{\mathrm{th}}+L_{\mathrm{nth}}$. In steady state, the source
luminosity is equal to the total injected power:
$L=L_{\mathrm{s}}+L_{\mathrm{h}}$. In the code, the luminosity is described by
the compactness parameter:

$l=\frac{\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c^{3}}\frac{L}{R},$ (1)

where $\sigma_{\mathrm{T}}$ is the Thomson cross-section, $m_{\mathrm{e}}$ the
electron mass, and $R$ is the typical source size. Other compactness
parameters $l_{s}$, $l_{h}=l_{th}+l_{nth}$ are defined in a similar manner.
The overall spectral shape is strongly dependent on the dominance of high-
energy emission, that is, the ratios $l_{\mathrm{h}}/l_{\mathrm{s}}$ and
$l_{\mathrm{nth}}/l_{\mathrm{h}}$.

####  3.3.2 Application to our data

As in all our different approaches, we freeze the system inclination to 27°.
In the LHS, we fix the temperature of the disk photons to $kT_{\mathrm{bb}}$
$=0.2$ keV and the disk compactness to $l_{\mathrm{s}}$ $=1$ (e.g., Gierliński
et al., 1999; McConnell et al., 2002; Del Santo et al., 2013). In the HSS, the
spectrum is dominated by the emission of the disk photons and therefore, we
let the disk temperature, $kT_{\mathrm{bb}}$, vary freely and fix
$l_{\mathrm{s}}$ to 10 (Gierliński et al., 1999; McConnell et al., 2002; Del
Santo et al., 2013). We further assume the Lorentz factor of the nonthermal
plasma to be power-law distributed between $\gamma_{\mathrm{min}}=1.3$ and
$\gamma_{\mathrm{max}}=1000$. The radius of the corona is set to $10^{7}$ cm
and we assume solar abundances ANGR (Anders & Tobin, 1989) for all metals in
the reflection modeling. The ionization parameter is set to zero and we do not
inject pairs. As in our previous approach, we also add a Gaussian to the HSS
model.

Table 3: eqpair model parameters for our stacked HSS and LHS spectra. Fixed parameters are denoted with an asterisk. Parameter | LHS | HSS
---|---|---
$c_{\mathrm{JEM-X}}$ | 1∗ | 1∗
$c_{\mathrm{ISGRI,\texttt{OSA10}}}$ | $0.96^{+0.02}_{-0.02}$ | –
$c_{\mathrm{ISGRI,\texttt{OSA11}}}$ | $0.99^{+0.04}_{-0.04}$ | $0.88^{+0.03}_{-0.03}$
$c_{\mathrm{SPI}}$ | $1.00^{+0.02}_{-0.02}$ | $1.20^{+0.00}_{-0.03}$
$kT_{\mathrm{bb}}$ [eV] | 200∗ | $565^{+68}_{-58}$
$l_{\mathrm{h}}/l_{\mathrm{s}}$ | $6.34^{+0.30}_{-0.10}$ | $0.42^{+0.05}_{-0.05}$
$l_{\mathrm{s}}$ | 1∗ | 10∗
$l_{\mathrm{nth}}/l_{\mathrm{h}}$ | $0.9^{+0.0}_{-0.2}$ | $>0.8$
$\tau_{\mathrm{p}}$ | $1.29^{+0.08}_{-0.08}$ | 0.2∗
$\Gamma_{\mathrm{inj}}$ | $2.9^{+0.2}_{-0.1}$ | $3.1^{+0.1}_{-0.4}$
$\Omega/2\pi$ | $0.25^{+0.01}_{-0.03}$ | $1.0^{+0.6}_{-0.2}$
$E_{\mathrm{Fe}}$ [keV] | – | 6.4∗
$\sigma_{\mathrm{Fe}}$ [keV] | – | 0.4∗
$\chi^{2}/\mathrm{dof}$ | $134.68/117$ | $101.81/79$

We show the best-fit model in Fig. 6 and list all parameters in Table 3. Our
parameters are consistent overall with the previous studies indicated above.

The LHS is clearly dominated by Comptonization at an optical depth of $\sim
1.3$ and with a high fraction of accelerated nonthermal particles with
$l_{\mathrm{nth}}/l_{\mathrm{h}}\sim 0.9$. This results in a spectrum that is
dominated by the hard X-rays, that is, $l_{\mathrm{h}}/l_{\mathrm{s}}\sim
6.3$. The accelerated electron distribution is a power law with an injection
electron index $\Gamma_{\mathrm{inj}}\simeq 2.9$ (Table 3).

Our HSS spectra show some emission up to 600 keV, i.e., to slightly higher
energies than in most previous studies. In this state, our fit becomes
unstable when letting $\tau_{\mathrm{p}}$ vary freely. We therefore freeze
this parameter to $\tau_{\mathrm{p}}$= 0.2 in line with the values previously
obtained with eqpair (Gierliński et al., 1999; Frontera et al., 2001;
McConnell et al., 2002; Sabatini et al., 2013; Del Santo et al., 2013) and in
coherence with our phenomenological approach (Sec. 3.2.2; Table 2). We note
that we need an injection electron index of $\Gamma_{\mathrm{inj}}\sim$ 3.0
consistent with the value we find in the LHS (Table 3). The spectrum is
dominated by the disk thermal emission as indicated by a
$l_{\mathrm{h}}/l_{\mathrm{s}}$ $\sim$0.4 fraction weaker and a photon disk
temperature of $kT_{\mathrm{bb}}$ $\sim$0.6 keV, which is higher than in the
LHS and is consistent with the results of Gierliński et al. (1999), Frontera
et al. (2001), McConnell et al. (2002), Sabatini et al. (2013), Del Santo et
al. (2013). The fraction of particles emitted by nonthermal processes
$l_{\mathrm{nth}}/l_{\mathrm{h}}$ is consistent with a purely nonthermal
distribution ($>77$ %). Finally, the reflected photon fraction is consistent
with our previous approach, although it is less constrained.

###  3.4 State-resolved spectral modeling: magnetized hybrid plasma

####  3.4.1 Basics of the belm model

In the previous section, we model the emission from an unmagnetized medium
with the eqpair model. It is nevertheless believed that the disk and the jet
are significantly magnetized (e.g., Blandford & Znajek, 1977; Ferreira et al.,
2006). Moreover, nonthermal particles are often assumed to be accelerated by
magnetic reconnection, which also requires strong magnetic fields (e.g.,
Poutanen & Vurm, 2009).

In this section we therefore extend the previous modeling by additionally
considering magnetic fields as a natural explanation for the accelerated,
nonthermal particle distribution that we used for eqpair. To do so, we use the
belm code (Belmont et al., 2008), which uses the same parameters as eqpair,
but additionally introduces the magnetic field intensity through the magnetic
compactness,

$\centering
l_{\mathrm{B}}=\frac{\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c^{2}}R\frac{B^{2}}{8\pi}.\@add@centering$
(2)

The inclusion of the magnetic field yields several effects over the modeling
presented so far. Firstly, additional synchrotron radiation is produced, which
can then be Comptonized (synchrotron self-Compton). This directly impacts the
spectral shape by adding a new broad-band component and indirectly by
increasing the particle cooling rate. Secondly, low-energy particles can
synchrotron-absorb high-energy photons, which heats these particles, resulting
in efficient thermalization.

To moderate the model degeneracy, we focus on a purely nonthermal model
($l_{\mathrm{th}}$ $=0$, see also Del Santo et al., 2013). Thus, the total
radiation compactness $l=l_{\mathrm{s}}+l_{\mathrm{th}}+l_{\mathrm{nth}}$
becomes $l=l_{\mathrm{s}}+l_{\mathrm{nth}}$, and
$l_{\mathrm{h}}=l_{\mathrm{nth}}$.

The table we use is computed with five model parameters: $l_{\mathrm{nth}}$,
$l_{\mathrm{B}}/l_{\mathrm{nth}}$, $\tau_{\mathrm{p}}$,
$\Gamma_{\mathrm{inj}}$ and $kT_{\mathrm{in}}$. As shown in Malzac & Belmont
(2009), the shape of the spectrum depends on ratios such as
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ or $l_{\mathrm{nth}}/l$ and not on the value
of the total compactness $l$. Therefore, its value is fixed to $l=10$ in the
table which represents the typical value of $0.01L_{\mathrm{Edd}}$ for both
states, while $l_{\mathrm{s}}$ is constrained by enforcing
$l_{\mathrm{s}}+l_{\mathrm{nth}}=10$. The exact magnetic field compactness
depends on the normalization $N$ of the model,

$l_{B,\mathrm{obs}}=\frac{l_{B}}{l_{\mathrm{nth}}}\frac{l_{\mathrm{nth}}}{l}l_{\mathrm{obs}}\quad\text{with}\quad
l_{\mathrm{obs}}=\frac{4\pi D^{2}\sigma_{T}}{Rc}Nl.$ (3)

The magnetic field intensity can then be estimated assuming the distance to
the source, $D=1.86$ kpc, and the size of the corona, $R=20\,R_{\mathrm{g}}$.
As previously, we fix the value of the injected seed photons to
$kT_{\mathrm{in}}=0.2$ keV in the LHS and $\tau_{\mathrm{p}}=0.2$ in the HSS.
An iron line fixed at 6.4 keV is also added in the HSS. Here, we describe
reflection using the convolution code reflect. In eqpair, the reflection model
ireflect is directly implemented as part of the model, with the additional
capability to model an ionized medium. In our case, we choose a neutral medium
in order to guarantee comparability between belm and eqpair.

####  3.4.2 Application to our data

Table 4 shows the best-fitting parameters for our modelings with belm. The
best-fit models and the residuals between those and the source spectra are
shown in Fig.6.

The LHS is only compatible with weak values of the magnetic compactness
parameter ($l_{\mathrm{B}}/l_{\mathrm{nth}}$ $<0.04$). This results in a
coronal magnetic field $B<4.83\times 10^{5}(20R_{\mathrm{g}}/R)$ G compatible
with the results obtained by Malzac & Belmont (2009), Del Santo et al. (2013),
and Poutanen & Vurm (2009). Larger fields produce large amounts of soft
synchrotron photons that Compton cool the thermal electron population. The
parameters obtained in this state (Table 4) are consistent with the parameters
found with the unmagnetized model eqpair (Table 3), except for the optical
depth for which we observe a larger value with belm. The difference in optical
depths may arise from a different treatment of the coronal geometry: belm uses
an escape probability corresponding to a pure spherical geometry, while eqpair
uses a combination of factors related to spherical and slab geometries.

In the HSS, we also find parameters consistent with the unmagnetized model.
The parameters are compatible with the previous results of Malzac & Belmont
(2009), Poutanen & Vurm (2009), Del Santo et al. (2013). We note that
$l_{\mathrm{B}}$ and $l_{\mathrm{s}}$ both contribute to the corona cooling.
$l_{\mathrm{s}}$ is by definition the total amount of cold photons entering in
the corona, and thus, the fraction of disk photons that participate in the
cooling is 100 %. We also test the possibility that a fraction of the disk
photons will not interact with the corona by adding a disk black body (diskbb)
in the model and investigate whether it has an effect on the value of
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ and/or $l_{\mathrm{nth}}$. The seed photon
temperature of diskbb is fixed to the temperature $kT_{\mathrm{in}}$ in the
belm model. We find that the uncertainties were slightly different:
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ $=1.6\pm 0.4$ and
$l_{\mathrm{nth}}=2.2^{+0.9}_{-0.2}$ with the addition of diskbb compared to
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ $=1.6^{+0.4}_{-0.7}$ and
$l_{\mathrm{nth}}=2.2^{+0.7}_{-0.2}$ without (Table 4). This latest value on
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ leads to a magnetic field of
$B=1.9^{+1.6}_{-1.3}\times 10^{6}(20R_{\mathrm{g}}/R$) G.

The shape of the Comptonization spectrum depends on the heating/cooling ratio;
a smaller $l_{\mathrm{s}}$ and a larger $l_{\mathrm{B}}$ can lead to the same
Comptonization spectrum. However, contrary to a disk, the thin synchrotron
spectrum extends above 1 keV; in the LHS, this component can be hidden by the
Comptonized component, but in the HSS, if there was a synchrotron emission
comparable to the disk emission, we should see it in the spectrum, which is
not the case and this is confirmed by the fits with an additional disk black-
body component. We discuss these results and their implications in the
following section.

Figure 6: Best models obtained from the fits to spectra for Cyg X-1 with the unmagnetized (left) and magnetized (right) hybrid corona models. Plain lines correspond to the full models whereas the loosely dashed line shows the iron line component. JEM-X, ISGRI, and SPI data are represented in light, intermediate, and darker colors, respectively. Table 4: belm model parameters for our stacked LHS and HSS spectra. Fixed parameters are denoted with an asterisk. Parameter | LHS | HSS
---|---|---
$c_{\mathrm{JEM-X}}$ | 1∗ | 1∗
$c_{\mathrm{ISGRI,\texttt{OSA10}}}$ | $0.96^{+0.03}_{-0.03}$ | –
$c_{\mathrm{ISGRI,\texttt{OSA11}}}$ | $0.99^{+0.03}_{-0.03}$ | $0.88^{+0.03}_{-0.03}$
$c_{\mathrm{SPI}}$ | $1.02^{+0.03}_{-0.03}$ | $1.20^{+0.00}_{-0.01}$
$kT_{\mathrm{in}}$ [eV] | 200∗ | $483^{+67}_{-26}$
$l_{\mathrm{nth}}$ | $9.0^{+0.6}_{-0.1}$ | $2.2^{+0.7}_{-0.2}$
$l_{\mathrm{B}}/l_{\mathrm{nth}}$ | $<0.04$ | $1.6^{+0.4}_{-0.7}$
$\tau_{\mathrm{p}}$ | $2.08_{-0.09}^{+0.08}$ | 0.2∗
$\Gamma_{\mathrm{inj}}$ | $3.0^{+0.2}_{-0.1}$ | $2.4^{+0.3}_{-0.4}$
$\Omega/2\pi$ | $0.16^{+0.03}_{-0.03}$ | $1.1^{+0.4}_{-0.2}$
$\chi^{2}/\mathrm{dof}$ | $141.61/117$ | $97.73/79$

##  4 Discussion

###  4.1 Summary of the main results

The extended INTEGRAL data set considered here allow us to confirm the
detection of a high-energy tail in the LHS (e.g., Cadolle Bel et al., 2006,
L11, J14, R15). We also report the detection of a high-energy tail in the HSS
with INTEGRAL, extending to at least 600 keV as seen with SPI (Fig. 4).

Our empirical model (Sect. 3.2) consists of purely thermal Comptonization
(compTT) with reflection off neutral gas and an independent power-law
component to model any high-energy tail emission. We bring solid constraints
for the high-energy tails in both states: with $\Gamma^{\mathrm{LHS}}\simeq
2.03$ and $\Gamma^{\mathrm{HSS}}\simeq 2.0$ (Table 2) they, in particular,
appear to have similar slopes. Their 200–1000 keV fluxes are $3.5\times
10^{-9}$ and $7.1\times 10^{-10}$ ergs/cm2/s which are somewhat different.

In an attempt to gain a more physical description of the data, we tested
hybrid Comptonization models for an unmagnetized (eqpair) and magnetized
(belm) corona. Our results are consistent with those of previous studies
(Gierliński et al., 1999; McConnell et al., 2002; Del Santo et al., 2013). The
fits with eqpair lead to a higher value of $l_{\mathrm{h}}/l_{\mathrm{s}}$ in
the LHS than in the HSS, which indicates that the source emission is dominated
by Comptonization in the LHS. Both states are well modeled and we obtain
electron injection indices of $\Gamma^{\mathrm{LHS}}_{\mathrm{inj}}\simeq 2.9$
and $\Gamma^{\mathrm{HSS}}_{\mathrm{inj}}\simeq 3.1$. These results are
globally consistent with the parameters we obtain with belm. Moreover, belm
provides us constraints on the magnetic field. We found an upper limit
$B<4.83\times 10^{5}(20R_{\mathrm{g}}/R)$ Gauss in the LHS, and we are able to
constrain $B=1.9^{+1.6}_{-1.3}\times 10^{6}(20R_{\mathrm{g}}/R)$ Gauss in the
HSS. Those values are consistent with upper limits found by Malzac & Belmont
(2009), Poutanen & Vurm (2009), Del Santo et al. (2013).

###  4.2 Interpretation: origin of the high-energy emission

The detection of a high-energy tail in both states leads us to consider two
different scenarios:

  * 1.

The high-energy tails in the LHS and HSS originate from the same medium.

  * 2.

The high-energy tails in the LHS and HSS are the signatures of different
physical mechanisms, and thus have distinct origins, despite having photon
indices of similar values.

####  4.2.1 Same origin and spectral interpretation

Our results with the semi-phenomenological approach are consistent with what
has been previously reported for Cyg X-1: a significantly higher optical depth
in the LHS compared to the HSS, which could suggest that the corona shrinks as
the system transits from the LHS to HSS (e.g., Zdziarski et al., 1998; Wilms
et al., 2006; Del Santo et al., 2013), and an electron temperature around 50
keV in the LHS. However, this approach does not allow us to treat the
broadband spectra in a self-consistent way, and prevents us from verifying
whether or not a single medium (a corona) can be responsible for the 100–1000
keV emission.

With the physical, broad-band hybrid spectral models eqpair and belm, in
contrast, we can study the evolutionary behavior of a corona as a single self-
consistent medium. The first conclusion we can draw is that our spectra are
well modelled with both the unmagnetized and magnetized models eqpair and
belm. In these frameworks, the high-energy emission is produced by a
nonthermal electron population in either a nonmagnetized or a magnetized
corona in both states.

As belm furnishes precious constraints on the magnetic field in both states,
we can, moreover, draw another conclusion which appears intriguing. The ratio
between the equipartition magnetic field compactness $l_{\mathrm{B_{R}}}$ and
the magnetic field compactness $l_{\mathrm{B}}$ is given by Lightman &
Zdziarski (1987):

$\frac{l_{\mathrm{B}}}{l_{\mathrm{B_{R}}}}=\frac{l_{\mathrm{B}}}{l_{\mathrm{nth}}}\frac{l_{\mathrm{nth}}}{l}\frac{4\pi/3}{1+\tau_{\mathrm{T}}/3}.$
(4)

We find $\frac{l_{\mathrm{B}}}{l_{\mathrm{B_{R}}}}<0.08$ and
$1.4^{+0.7}_{-0.8}$ for the LHS and HSS, respectively. In the LHS, this low
value argues against magnetic reconnection powering the corona, which is
consistent with the conclusions of Malzac & Belmont (2009), Poutanen & Vurm
(2009), and Del Santo et al. (2013). If electron acceleration is not possible
via magnetic reconnection, it thus implies that the high-energy emission must
be produced in a second, independent zone, which is not the corona, and could
for example be the jets.

In the framework of belm as a one-zone model, the MeV tail is mainly due to
Comptonization and would be inconsistent with the observed high degree of
polarization claimed for the LHS so far (L11, Jourdain et al., 2012, R15). In
the HSS, the magnetic field is large enough to be able to produce magnetic
reconnection, and this thus could be the mechanism responsible for the
electron acceleration in the corona (Malzac & Belmont, 2009; Poutanen & Vurm,
2009; Del Santo et al., 2013). To summarize, an unmagnetized corona could be
at the origin of the high-energy emission in both states and is thus a
plausible scenario for both, while a magnetized corona seems to be working
only in the HSS.

####  4.2.2 Different origin and comparison with previous results

As indicated, the tails in both long-lasting spectral states do not
necessarily have the same origin. Evolved radio jets (e.g., Rushton et al.,
2012, and references therein), the high polarization fraction of the high-
energy tail above $400$ keV (L11, Jourdain et al., 2012, R15), and broadband
high-energy spectral features associated with the jets of Cyg X-1 (L11,
Jourdain et al. 2012, Zdziarski et al. 2012, R15, Walter & Xu 2017) seem to
point towards a synchrotron jet emission in the LHS. Several studies
investigate this possibility. Zdziarski et al. (2012) apply their jet model
and find that such emission requires an electron acceleration at a rather hard
power-law index (1.3–1.6), which is difficult to reconcile with standard
acceleration mechanisms. Similar conclusions were found in an another source
observed with INTEGRAL, namely GRS 1716–246, for which Bassi et al. (2020)
used a jet internal shock emission model (Malzac, 2013) in order to model the
high-energy emission of this source. However, another study on Cyg X-1 shows
that the emission during the LHS can be explained by a self-consistent jet
model (Kantzas et al., 2020). In their model, a broadband jet-synchrotron
component is the origin of both the radio continuum and the MeV tail in the
LHS. A photon index of $\Gamma\sim 2$ in the MeV range would be compatible
with their results. The jet origin is further supported by the detection
between 60 MeV and 20 GeV by Fermi/LAT (Bodaghee et al., 2013; Malyshev et
al., 2013; Zanin et al., 2016; Zdziarski et al., 2017), which may also hint
towards a hadronic contribution to the jet-accelerated plasma (Kantzas et al.,
2020).

A scenario that lies beyond the scope of our paper, and is hard to constrain
and fully discuss with our data, is the putative presence of a jet in both
states. This possibility arises from recent claims of possible detected jets
in the HSS. In particular, shortly after the LHS-to-HSS state transition
around MJS 55400, Rushton et al. (2012) detect a weak and compact jet on VLBI
scales. It is unclear whether this feature has remained undetected in the
long-term stable HSS before, or whether this is a remnant feature of the state
transition that happened close-by in time. A recent study claims the detection
of radio jets in the HSS of Cyg X-1 (Zdziarski et al., 2020). The authors
observe a clear correlation between 15 GHz radio flux and the simultaneous
15–50 keV flux not only in the LHS but also, albeit with a different slope, in
the HSS. However, as opposed to the LHS, the soft X-rays at a few keV are
reported to be time-lagged by about 100 days in the HSS, suggesting advection
from the donor through the accretion disk. The absence of a time-lag between
the radio and hard (15–50 keV) X-ray emission in both states might, here,
imply a jet origin for the latter hard X-ray radiation. However, we note that
the significantly lower degree of polarization below $\sim 400$ keV (Jourdain
et al., 2012) and both our spectral modeling and that of Kantzas et al. (2020)
suggest that the persistent jet may only contribute above hundreds of keV
during the LHS.

No radio emission has been detected in any other microquasar in the HSS down
to a very low flux level (e.g., Fender et al., 1999; Remillard & McClintock,
2006; Belloni, 2010). Still, a recent study suggests that jets could survive
in the HSS with heavily suppressed radio emission that may or may not be
detectable depending on the observing technique and intrinsic radio-flux
level. These so-called “dark jets” are kinematically dominated and radiatively
inefficient (Drappeau et al., 2017). In the context of the internal shock
model of Drappeau et al. (2017), the radio flux is quenched, but a kinematic
jet driven by the disk rather than the corona still subsists. However, this
study is based on a broadband modeling of GX 339$-$4 and H 1743$-$322, both of
which are microquasars with low-mass companion stars and are purely Roche lobe
accreting. It is unclear whether or not the results from Drappeau et al.
(2017) can be applied to Cyg X-1, a persistently wind-accreting system with a
high-mass companion.

##  5 Conclusion

In this paper, we advance the long-lasting investigation of the nature of the
hard MeV tail of Cyg X-1 for both major spectral states. We accumulate data
from 15 years of INTEGRAL (JEM-X, IBIS and SPI) observations of the source and
reach unprecedented count statistics in the 3-2000 keV range, not only in the
LHS, but also in the HSS, which allows us to firmly confirm the suspected
presence of a high-energy component —extending to at least 600 keV– in the
spectra of the source.

We follow two approaches with the modeling of these data. In the first,
empirical model, we fit a thermal Comptonization spectrum independent of a
simple power law, describing the MeV tail between hundreds of keV and $\sim$1
MeV. The slopes of the MeV tails appear compatible between the LHS and the
HSS. The origin of this high-energy emission can be tackled with single
physical spectral models where the data are self-consistently considered. We
therefore used the hybrid thermal/nonthermal Comptonization codes eqpair and
belm, accounting for an unmagnetized and magnetized single-zone corona,
respectively. We reasonably successfully reproduce the spectral changes across
the states of Cyg X-1 overall. Within this single-zone approach, an
unmagnetized corona is able to account for the high-energy emission in both
states. Considering the magnetized coronal model, in the LHS, the magnetic
field is too weak to be able to accelerate electrons via magnetic
reconnection. In the HSS, this mechanism could be responsible for the
nonthermal emission we observe (Malzac & Belmont, 2009; Poutanen & Vurm, 2009;
Del Santo et al., 2013).

We then also discuss our results in the context of physical broadband multi-
zone models from the literature and their ability to reproduce the radio-to-
gamma-ray emission of Cyg X-1. In the LHS, a persistent radio jet can explain
both the radio emission and the MeV tail via a direct synchrotron component,
while the X-rays at intermediate energies are well described with hybrid
Comptonization. When switching to the HSS, we suggest that the persistent jet
may transition to a quenched or failed jet, leaving behind a hybrid corona
that could either be magnetized or not, and which is responsible of the
nonthermal emission.

Our study does not provide conclusive evidence in favor of any particular one
of the scenarios proposed to explain the origin of the high-energy tail in the
HSS. A polarimetric analysis of the MeV tails for both states is in
preparation, extending the work previously presented in R15. Polarimetry will
be key to breaking degeneracies arising between the different physical models,
but is currently still affected by large uncertainties. Continued observations
and further studies of Cyg X-1, especially in the HSS at energies beyond 1 MeV
with INTEGRAL, will be crucial to increasing the count statistics and thus to
better constraining the slope of the MeV tail. Complementary studies of other
sources would facilitate our understanding of universal properties and
differences in such high-energy tails beyond Cyg X-1.

###### Acknowledgements.

We thank D. Kantzas, M. Lucchini, A. Chhotray, and S. Markoff for extensive
discussions on the origin of the high-energy tail emission based on physical
jet models developed at the Anton Pannekoek Institute for Astronomy, Univ. of
Amsterdam. FC, JR, PL acknowledge partial funding from the French Space Agency
(CNES), and the French programme national des hautes énergies (PNHE). FC and
VG acknowledge support from the ESTEC Faculty Visiting Scientist Programme. TS
is supported by the German Research Society (DFG-Forschungsstipendium SI
2502/1-1). VG is supported through the Margarete von Wrangell fellowship by
the ESF and the Ministry of Science, Research and the Arts Baden-Württemberg.
This work was partly funded by Deutsches Zentrum für Luft- und Raumfahrt grant
50 OR 1411 and 50 OR 1909. We made use of ISIS functions provided by
ECAP/Remeis Observatory and MIT (http://www.sternwarte.uni-erlangen.de/isis/).
We thank J.E. Davis for the development of the slxfig module that has been
used to prepare the figures in this work. This work is based on observations
with INTEGRAL, an ESA project with instruments and science data centre funded
by ESA member states (especially the PI countries: Denmark, France, Germany,
Italy, Switzerland, Spain), and with the participation of the Russian
Federation and the USA. This research has made use of NASA’s Astrophysics Data
System Bibliographic Services (ADS).

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##  Appendix A Variability timescales in Cyg X-1: the AIC map (SPI)

Figure 7: Example of AIC maps with a different number of parameters for the
background and the source variabilities.

Figure 7 shows AIC values as a result of maximum likelihood fits with
different numbers of parameters for the background variability (timescale: the
more parameters, the more variable) and the source (Cyg X-1) variability
timescale, which was carried out in order to determine the flux and its
uncertainty in each bin. In the HSS, the AIC maps appear with a clear minimum
(most probable solution according to AIC), slowly changing the source
variability from $\sim$10 h to constant as a function of energy. In the LHS,
the AIC minimum is found at the smallest timescale possible (one scw) for both
source and background. For higher energies, the background still appears to
vary on the shortest possible timescale, and the source seems constant. This
appears unreasonable as the background variability should not change as a
function of energy. We believe that the LHS spectral extraction suffers from a
degeneracy in the determination of the number of required parameters. We
therefore “calibrate” our spectrum using the IBIS spectral extraction for the
LHS: using a source model without temporal variability but allowing the
background to freely vary leads to an absolute normalization and shape close
to the IBIS extraction.

##  Appendix B Comparison of different SPI spectral extraction methods

Here, we present two other spectral extraction methods for the SPI data.

###  B.1 Method 2: Using different event selection

In this method, we consider the threshold of the PSD events, as set in the SPI
telescope electronics. These are from 400 keV to 2.2 MeV after revolution 1170
and from nearly 650 keV to 2.2 MeV before. In this energy range, we used the
PSD events only. This selection has been prescribed in Roques & Jourdain
(2019) in order to avoid spectral artifacts caused by potential spurious
events. These spurious events appear at high energies ($>400$ keV) for very
bright sources (typically above 1 Crab). They are caused by a displacement of
low-energy photons toward high energy, when the source is so bright that many
photons can be detected in the same detection electronics time window. The
energy reconstruction for the SE chain is then not reliable. This does not
happen in the PE chain, which records the same event much more rapidly.

###  B.2 Method 3: Using SPIDAI

In this method, we use the SPI Data Analysis
Interface666https://sigma-2.cesr.fr/integral/spidai. (SPIDAI) to reduce the
data from SPI. We created a sky model containing Cyg X-1 and Cygnus X-3 and
set the source variability to ten scws for Cygnus X-3 and five scws for Cyg
X-1 (e.g., Bouchet et al. 2003; Cangemi et al. 2021). We then created the
background model by setting the variability timescale of the normalization of
the background pattern to five scws. In order to avoid the effects due to
solar flares, radiation belt entries, and other nonthermal incidents, we
removed scws for which the reconstructed counts compared to the detector
counts give a $\chi^{2}>1.5$. This selection reduces the total number of scws
by $\sim 10$ %. The shadowgrams were then deconvolved to obtain the source
flux, and spectra were extracted between 20 and 2000 keV using 27
logaritmically spaced channels.

Given the large number of scws in both states, we extracted six and four
different spectra for the LHS and HSS, respectively, each of them was
extracted from data of 500-600 scws and corresponds to a distinct period. For
each spectrum, we applied a correcting factor of 1/0.85 (above 400 keV or 650
keV according to the considered period, Roques & Jourdain 2019) to account for
the change efficiency. We then computed two average spectra for both states.
In the HSS, the normalization of the spectrum has a slight variability over
the four different periods which explain the larger error bars that we observe
in this state when computing the average spectrum.

###  B.3 Comparison between the methods

The right pannel on Fig. 8 shows the broad SPI spectra (20–2000 keV) used in
this paper along with the spectra extracted with the aforementioned methods.
The left pannel on Fig. 8 shows the same spectra between 400 keV and 700 keV,
that is to say the range for which the selection between SEs and PSD events is
different from the the method described in this paper and method 2, described
in Sec. B.1.

In Fig. 8, we show the residuals we obtain by fitting the spectra with a
simple cutoff powerlaw (constant*cutoffpl, $\chi^{2}_{\mathrm{LHS}}=75.50/76$,
$\chi^{2}_{\mathrm{HSS}}=93.65/76$). By fixing the constant parameter of the
spectra used in this paper to 1, we obtain constant values of
$c_{\mathrm{method\ 2}}=0.97\pm 0.02$ and $c_{\mathrm{method\ 3}}=1.06\pm
0.03$ for the LHS and $c_{\mathrm{method\ 2}}=1.00\pm 0.02$ and
$c_{\mathrm{method\ 3}}=0.86\pm 0.06$ for the HSS. For both states, the three
spectra are consistent and we do not observe any significant differences, in
particular in the 400–650 keV range, the range most affected by spectral
artifacts caused by potential spurious events.

We also investigate whether the parameters extracted in this study would be
different using method 2 or method 3 spectrum. We find that the parameters
remain unchanged and zre consistent within the uncertainties for both states
when applying the semi-physical model or the hybrid corona models (eqpair and
belm).

Figure 8: Left pannel: Spectra extracted with three different methods: method
1 described in Sec. 2.3 of this paper (dark red), method 2 described in Sec.
B.1 (medium red), and method 3 using SPIDAI (light red). Right pannel: Focus
on the 400-700 keV energy range.