\section{Introduction}
\label{intro}

Compton scattering of soft seed photons has for long been shown to explain well hard X-ray spectra of the accreting black-hole (BH) X-ray binaries (XRBs) in their hard spectral state. The electron distribution appears to be predominantly thermal at mildly relativistic temperatures, $kT_{\rm e}$, see, e.g., \citet{DGK07} and \citet{Z20_thcomp} and references in those works. However, the form of the spectral high-energy cutoff in some sources indicates the presence of a substantial high-energy non-thermal tail beyond a Maxwellian electron distribution, e.g., \citet{McConnell02, Wardzinski02, PV09, ZMC17, Walter17}. On the other hand, instead of being thermal, the dominant Comptonization process could be scattering on the bulk motion of fast-moving plasmoids generated in magnetic reconnection, in which case a photon tail due to electron acceleration can appear as well \citep{Beloborodov17, Sironi20}.

A related effect is e$^\pm$ pair production in photon-photon collisions. High-energy tails in the spectra of sources explained by hybrid Comptonization usually cross the threshold for this process, $m_{\rm e}c^2$, significantly increasing the number of pair-producing photons with respect to the purely thermal case. Pair equilibria (in which the rate of pair production is balanced by that of pair annihilation) in hybrid plasmas were studied in \citet{ZLM93}, \citet{Coppi99}, \citet{G99}, \citet{McConnell02}, \citet{MB09} and \citet{Fabian17}.

Here we study these effects in {MAXI J1820+070}\xspace. It is a transient BH XRB, whose outburst was discovered in 2018 \citep{Kawamuro18}. The source is relatively nearby, at an accurately measured distance of $d\approx 3.0\pm 0.3$\,kpc \citep{Atri20}.  Also, the inclinations of both the binary and its radio jet are well constrained, as $i_{\rm b}\approx 66\degr$--$81\degr$ \citep{Torres20}, $i_{\rm j}\approx 63\degr\pm 3\degr$ \citep{Atri20}, respectively. The BH mass is given by $M\approx (5.95\pm 0.22){\rm M}_{\sun}/\sin^3 i_{\rm b}$ \citep{Torres20}. The outburst was extensively monitored by a number of observatories, in particular by {\it Nuclear Spectroscopic Telescope Array} ({\textit{NuSTAR}}\xspace; \citealt{Harrison13}), the Spectrometer on {\textit{INTEGRAL}}\xspace (\citealt{Roques03}; SPI), and the Imager on Board the {\textit{INTEGRAL}}\xspace Satellite (\citealt{Ubertini03}; IBIS). Here, we study spectra from contemporaneous observations by those instruments during the initial hard state of the outburst. They provide high-quality broad-band spectra in the range from 3\,keV up to $\gtrsim$1\,MeV.

\section{Observations and data reduction}
\label{data}

\renewcommand{\arraystretch}{0.9}\begin{table*}\centering
\caption{Contemporaneous observations of {MAXI J1820+070}\xspace with {\textit{INTEGRAL}}\xspace and {\textit{NuSTAR}}\xspace in the hard state
}
\vskip -0.4cm
\begin{tabular}{lcccccccccc}
\hline
Epoch & {\textit{INTEGRAL}}\xspace & SPI Start time & Exposure  & IBIS Start time & Exp.\ ISGRI &{\textit{NuSTAR}}\xspace Obs. ID & Start time & Exposure A \\
& revolution& End time && End time &Exp.\ PICSiT && End time&  Exposure B \\
\hline
1 & 1934 &58201.555& 13604 &58201.544 &8796 &90401309008 &58201.526&3046  \\
&&58201.757&&58201.757&&&58201.766 &3214\\
2 & 1938 &58212.181 &28507 &58212.181 &8305 &90401309012 &58212.200&12334   \\
&&58212.606&&58212.393&&&58213.177& 12964\\
A &1931--1951 &58193.453& 1147756 &58193.455 &765345& 90401309004--16 & 58198.036& 50176\\
&&58246.894&&58246.892 & 1156304 &&58242.643& 52799\\
\hline
\end{tabular}
\tablecomments{
The times are given in MJD, and the exposures are effective in seconds. The average {\textit{NuSTAR}}\xspace spectrum is from 7 observations with the Obs.\ IDs 904013090(04, 06, 08, 10, 12, 14, 16).}
\label{log}
\end{table*}

We use the {\textit{INTEGRAL}}\xspace data from SPI and IBIS. The latter consists of the ISGRI and PICSiT detectors. We use the SPI data as published in \citet{Roques19}. Data reduction and spectral extraction for ISGRI was done using the OSA v.11.1 software \citep{Courvoisier03}. The analysis of the PICSiT data follows the method of \citet{Lubinski09}.

We also use data from {\textit{NuSTAR}}\xspace. They were reduced with {\sc heasoft} v.6.25, the {\tt NUSTARDAS} pipeline v.1.8.0, and {\tt CALDB} v.20200912. We set {\tt saamode=strict}, {\tt tentacle=yes} and {\tt STATUS== b0000xxx00xxxx000}. The source region is a $60''$ circle centered on the peak brightness. We group the data to signal-to-noise ratio $\geq$50, but only below 69\,keV, so we can utilize the 3--79\,keV band to the maximum.

The chosen data sets are listed in Table \ref{log}. Epochs 1 and 2 consist of overlapping {\textit{INTEGRAL}}\xspace and {\textit{NuSTAR}}\xspace observations. They correspond to the beginning of a plateu phase on the count-rate/hardness plot, see fig.\ 2 in \citet{Buisson19}. For them, the PICSiT data have low statistics and we do not use them. Then, we analyze the average {\textit{INTEGRAL}}\xspace spectra for the initial hard state, over the span of 53 d, and the corresponding average {\textit{NuSTAR}}\xspace spectra, which set is denoted as A in Table \ref{log}. These {\textit{INTEGRAL}}\xspace observations start in the middle of the initial sharp rise phase of the outburst and continue through the middle of the plateau phase. This data set includes also the average PICSiT spectrum.

We use the SPI spectra from 23\,keV up to 650\,keV for epochs 1, 2, and up to 2.2\,MeV for the average spectrum, and include in them a 0.5\% systematic error (\citealt{Roques19}; added in quadrature). We add the same systematic error to the average {\textit{NuSTAR}}\xspace spectra, in order to avoid the broad-band fit to be dominated by the very good statistics of the latter. However, following previous papers on the {\textit{NuSTAR}}\xspace data from {MAXI J1820+070}\xspace, e.g., \citet{Buisson19}, we do not add a systematic error to the individual spectra. Given apparent calibration inaccuracies in the ISGRI and PICSiT data, we add a 1\% systematic error to each of them. Also, the ISGRI fluxes at its lowest energies are substantially below those of the other detectors, and thus are used in the 32--500\,keV range only. We use the PICSiT data in the 0.24--2\,MeV range.

\section{Spectral Fits}
\label{fits}

\subsection{The method}
\label{method}

In our spectral analysis, we follow the treatment of \citet{Zdziarski21b} ({\citetalias{Zdziarski21b}}\xspace) with some modifications related to the presence of high-energy non-thermal tails in the studied spectra. We use the X-ray fitting package {\sc{xspec}} \citep{Arnaud96}. The fit uncertainties are for 90\% confidence, $\Delta\chi^2 \approx 2.71$. Residual differences between the calibration of the {\textit{NuSTAR}}\xspace Focal Plane Modules A and B and the {\textit{INTEGRAL}}\xspace detectors SPI, ISGRI and PICSiT are accounted for by the model {\tt plabs}, which multiplies the model spectra by $K E^{-\Delta\Gamma}$. We set $K$ and $\Delta\Gamma$ fixed at 1 and 0, respectively, for the {\textit{NuSTAR}}\xspace A module, and find them fitted to the module B as 0.99 and $\approx +0.01$, respectively. For the {\textit{INTEGRAL}}\xspace epochs 1, 2, we find $|\Delta\Gamma|<0.015$ and $1.2<K<1.3$. We account for the ISM absorption using {\tt tbabs} \citep{WAMC00} using the elemental abundances of \citet{AG89}, with the column density toward the source of $N_{\rm H}=1.4\times 10^{21}$\,cm$^{-2}$ (e.g., \citealt{Kajava19}). At this low value, absorption of photons at $E\geq 3$\,keV is weak, but still noticeable.

We model spectra from Comptonization going directly to the observer and those reflected from an accretion disk around a Kerr BH taking into account atomic and relativistic effects. We use the {\tt reflkerr} \citep{Niedzwiecki19} and {\tt reflkerr\_bb} routines. The former assumes the Comptonization is on Maxwellian electrons, using the {\tt compps} code \citep{PS96}. The {\tt reflkerr\_bb} is a new routine\footnote{See \url{users.camk.edu.pl/mitsza/reflkerr}. It can also account for blackbody-like disk emission due to its irradiation and viscous disk dissipation, analogously to the lamppost version {\tt reflkerr\_lpbb} \citep{Zdziarski21}; however, we do not use that feature in this work.}, which allows for the presence of a power-law tail in the distribution, also using {\tt compps}. The tail is parametrized by the Lorentz factor ($\gamma_{\rm min}$) at which the electron distribution switches from thermal to a power law with an index, $p$, ${\rm d}N_{\rm e}/{\rm d}\gamma\propto \gamma^{-p}$, up to a maximum Lorentz factor, $\gamma_{\rm max}$, assumed to be $10^3$.

The thermal electrons are parametrized by their temperature, $kT_{\rm e}$, and the Compton parameter, $y\equiv 4\tau_{\rm T} kT_{\rm e}/m_{\rm e}c^2$, where $\tau_{\rm T}$ is the Thomson optical depth of the plasma. The Comptonizing cloud is assumed to be spherical and we set ${\tt geom}=0$ in {\tt compps}, which gives a fast but approximate method; in particular, we found it overestimates $y$. We therefore use an option of ${\tt geom}=-5$ (for a sinusoidal distribution of the seed photons in a sphere, \citealt{ST80}), which calculation is very slow but accurate, in order to find the actual values of $y$, and thus of $\tau_{\rm T}$. In addition, we evaluate the average 2--10\,keV power-law spectral index, $\Gamma$ of a fitted Comptonization component (unabsorbed and without reflection). The seed photons for Comptonization are assumed to have a blackbody distribution with the temperature of $kT_{\rm bb}=0.2$\,keV, following the results of the fits to the hard-state spectra from the Neutron star Interior Composition ExploreR (NICER\xspace; \citealt{Gendreau16}) of \citet{Wang20_HXMT}.

In {\tt reflkerr}, the reflected spectra in the rest frame are calculated using {\tt xillverCp} (v.\ 1.4.3; \citealt{GK10,Garcia18}) up to $\approx$10\,keV and {\tt ireflect} \citep{MZ95} at higher energies, see \citet{Niedzwiecki19} for details. Both the direct and reflected components are then integrated over the surface of a Keplerian disk taking into account the relativistic effects. Here, we assume the standard profile of $\propto R^{-3}$, which is the same as the disk viscous dissipation at $R\gg R_{\rm ISCO}$, where $R_{\rm ISCO}$ is the radius of the innermost stable circular orbit. The reflection fraction, ${\cal R}$, is defined in {\tt reflkerr} as the ratio of the flux irradiating the disk to that emitted outside in a local frame. We assume a rotating BH with the dimensionless spin of $a_*=0.998$, for which $R_{\rm{ISCO}}\approx 1.237 R_{\rm{g}}$, where $R_{\rm g}\equiv GM/c^2$. However, the effect of this assumption is negligible at $R\gg R_{\rm{ISCO}}$.

We then use the same geometrical model of the accretion flow as {\citetalias{Zdziarski21b}}\xspace, in which there are two Comptonization components, soft and hard, which reflect from two zones in the disk, as shown in fig.\ 4 of {\citetalias{Zdziarski21b}}\xspace. The soft component forms a corona covering the disk between its inner truncation radius, $R_{\rm in}>R_{\rm{ISCO}}$, up to the transition radius, $R_{\rm tr}\equiv R_{\rm in}+\Delta R$, and its reflection is from the disk underneath. This is compatible with the hard component forming a hot flow or a slow jet sheath with a large scale height at radii $<R_{\rm in}$. Since the disk between $R_{\rm in}$ and $R_{\rm tr}$ is covered by the soft corona, the reflection of the hard Comptonization component is from remote parts of the disk, at $R>R_{\rm tr}$. The fitted parameters are $R_{\rm in}$ and $\Delta R$, determined predominantly by the reflection of the soft and hard incident spectra, respectively. Each of the two reflection zones is characterized by its ionization parameter, $\xi\equiv 4\pi F_{\rm irr}/n$, where $F_{\rm irr}$ is the irradiating flux and $n$ is the density of the reflecting part of the disk.

\renewcommand{\arraystretch}{0.9}
\begin{table*}
\caption{The results of spectral fitting for our two-component coronal model (soft thermal and hard hybrid)
}
   \centering\begin{tabular}{ccccc}
\hline
Component & Parameter & Epoch 1 & Epoch 2 & Average\\
\hline
ISM absorption & $N_{\rm H}$ $[10^{21}]$\,cm$^{-2}$ & \multicolumn{3}{c}{1.4f}\\
\hline
Joint constraints & $i$ $[\degr$] & $70^{+1}_{-2}$ & $60^{+1}_{-1}$ & $66^{+1}_{-1}$\\
& $Z_{\rm Fe}\, [\sun]$ & $1.4^{+0.3}_{-0.2}$ & $1.2^{+0.1}_{-0.1}$ & $2.0^{+0.1}_{-0.2}$\\
\hline
Thermal Comptonization  & $y_{\rm th}$ & $0.55^{+0.02}_{-0.02}$ & $0.54^{+0.01}_{-0.02}$ & $0.55^{+0.01}_{-0.01}$\\
and reflection  &$\Gamma_{\rm th}$ & $1.82^{+0.03}_{-0.01}$ & $1.82^{+0.01}_{-0.01}$ & $1.85^{+0.02}_{-0.03}$\\
& $kT_{\rm e,th}$ [keV] & $12^{+1}_{-11}$ & $13^{+1}_{-1}$ & $12^{+1}_{-1}$\\
&$R_{\rm in}\, [R_{\rm g}]$ & $22^{+6}_{-6}$ & $27^{+6}_{-3}$ & $103^{+16}_{-15}$\\
& ${\cal R}_{\rm th}$ & $0.68^{+0.20}_{-0.15}$ & $0.36^{+0.02}_{-0.03}$ & $0.33^{+0.01}_{-0.02}$\\
& $\log_{10} \xi_{\rm th}$ & $3.61^{+0.07}_{-0.06}$ & $3.46^{+0.02}_{-0.03}$ & $3.65^{+0.01}_{-0.02}$\\
& $N_{\rm th}$ & $2.69^{+0.12}_{-0.17}$ & $2.83^{+0.52}_{-0.06}$ & $3.28^{+0.02}_{-0.03}$\\
\hline
Hybrid Comptonization &$y_{\rm h}$ & $0.95^{+0.07}_{-0.03}$ & $0.85^{+0.06}_{-0.02}$ & $0.78^{+0.01}_{-0.01}$\\
and reflection &$\Gamma_{\rm h}$  & $1.39^{+0.02}_{-0.06}$ & $1.40^{+0.01}_{-0.05}$ & $1.28^{+0.01}_{-0.01}$\\
& $kT_{\rm e,h}$ [keV] & $22.1^{+0.7}_{-0.4}$ & $21.3^{+0.3}_{-0.7}$ & $16.4^{+0.1}_{-0.7}$\\
&$\gamma_{\rm min}$ & $1.32^{+0.01}_{-0.02}$ & $1.27^{+0.01}_{-0.01}$ & $1.17^{+0.01}_{-0.01}$\\
&$p$ & $3.61^{+0.37}_{-0.11}$ & $4.15^{+0.16}_{-0.12}$ & $5.55^{+0.07}_{-0.05}$\\
&$\Delta R\, [R_{\rm g}]$ & $880^{+32}_{-594}$ & $331^{+143}_{-114}$ & $4^{+39}_{-3}$\\
& ${\cal R}_{\rm h}$ & $0.39^{+0.05}_{-0.01}$ & $0.24^{+0.02}_{-0.02}$ & $0.21^{+0.01}_{-0.02}$\\
& $\log_{10} \xi_{\rm h}$ & $2.47^{+0.11}_{-0.08}$ & $0^{+1.07}$ & $1.3^{+0.3}_{-1.3}$\\
& $N_{\rm h}$ & $1.34^{+0.32}_{-0.09}$ & $1.28^{+0.24}_{-0.04}$ & $0.80^{+0.01}_{-0.04}$\\
\hline
& $\chi_\nu^2$  & 953/792 & 1750/1307 & 2255/2011\\
\hline
\end{tabular}
\tablecomments{
The fitted model is {\tt plabs*tbabs(reflkerr+reflkerr\_bb)}. $Z_{\rm Fe}$ is the Fe abundance in solar units, $R_{\rm tr}=R_{\rm in}+\Delta R$, $R_{\rm out}=10^3 R_{\rm g}$, $N_{\rm th,h}$ is the flux density of a Comptonization component at 1\,keV in the observer's frame, $y_{\rm th,h}$ is the Compton parameter, $\Gamma_{\rm th,h}$ is the power-law index fitted to a given component in the 2--10\,keV range (thus not a free parameter), and ${\cal R}_{\rm th,h}$ is the reflection fraction. }
\label{t_fits}
\end{table*}

\subsection{Fit results}
\label{results}

\begin{figure*}[t!]
\centerline{
\includegraphics[width=5.8cm]{eeuf_epoch1_th-eps-converted-to.pdf}
\includegraphics[width=5.8cm]{eeuf_epoch2_th-eps-converted-to.pdf}
\includegraphics[width=6.6cm]{eeuf_epochA_th-eps-converted-to.pdf}}
  \caption{The {\textit{NuSTAR}}\xspace (magenta and cyan), SPI (blue) and ISGRI (red) unfolded spectra of epochs (a) 1, (b) 2, (c) A, fitted in the 3--80\,keV range with the two-component coronal model with thermal electrons (black solid curves), but shown in the entire ranges of the data. The spectra are normalized to the {\textit{NuSTAR}}\xspace A module. The fits significantly underpredict the spectra at $\gtrsim$100\,keV. The softer and harder Comptonization components are shown by  the red long dashes and blue short dashes, and the corresponding reflection components are shown by the magenta dot-dashes and cyan dots, respectively.
}\label{eeuf_th}
\end{figure*}

\begin{figure*}[t!]
\centerline{
\includegraphics[width=5.7cm]{eeuf_ratio_epoch1-eps-converted-to.pdf}
\includegraphics[width=5.7cm]{eeuf_ratio_epoch2-eps-converted-to.pdf}
\includegraphics[width=6.45cm]{eeuf_ratio_epochA-eps-converted-to.pdf}}
  \caption{The {\textit{NuSTAR}}\xspace (magenta and cyan), SPI (blue), ISGRI (red) and PICSiT (black; for A only) unfolded spectra and data-to-model ratios of the epochs (a) 1, (b) 2, (c) A, fitted in the full ranges with the two-component coronal model (black solid curves), see Table \ref{t_fits}. The spectra are normalized to the {\textit{NuSTAR}}\xspace A one. The softer (red long dashes) and harder (blue short dashes) Comptonization components are for thermal and hybrid electrons, respectively. The corresponding reflection components are shown by the magenta dot-dashes and cyan dots, respectively.
}\label{eeuf_ratio}
\end{figure*}

We first fit the {\textit{NuSTAR}}\xspace data alone. Initially, we assume that the two Comptonization zones have the same electron temperature. We obtain results similar to those in table 2 in {\citetalias{Zdziarski21b}}\xspace, with $R_{\rm in}\gg R_{\rm ISCO}$. The inclination is obtained from the reflection spectroscopy, and it is within $i\approx 60$--$70\degr$, which agrees with the binary and jet measurements (Section \ref{intro}). As in {\citetalias{Zdziarski21b}}\xspace, we do not impose the same inclination for each data set, in order to demonstrate that each of them gives $i$ compatible with the observational constraints, as well as to account for possible flaring and precession of the disk. We then include the SPI and ISGRI data and fit them simultaneously with {\textit{NuSTAR}}\xspace up to 80\,keV. We find an excellent agreement of those data with {\textit{NuSTAR}}\xspace (except for the their higher overall normalization, $K\approx 1.25$, and $\Delta\Gamma\approx +0.01$). In the next step, we include the data $>$80\,keV, and plot the resulting $EF_E$ spectra in Figure \ref{eeuf_th} (where $F_E$ is the differential energy flux). We find that both the SPI and ISGRI data at $E\gtrsim 100$\,keV are significantly above the model fitted at 3--80\,keV, while they are in a good mutual agreement. In particular, the spectra measured by the SPI and ISGRI extend up to 400\,keV with significant non-zero fluxes, while the models fitted in the 3--80\,keV range predict only very weak such emission.

These results show that fitting the {\textit{NuSTAR}}\xspace data with thermal Comptonization in order to determine the rate of e$^\pm$ pair production and constrain its equilibrium with the rate of pair annihilation in accreting BHs \citep{Fabian15} may be unreliable. At least in {MAXI J1820+070}\xspace, the {\textit{NuSTAR}}\xspace spectra fitted by models with a thermal electron distribution do not provide reliable predictions for the pair production rate. This conclusion agrees with that of \citet{Coppi99} and \citet{Fabian17}, who found that allowing for a hybrid electron distribution reduces the minimum electron temperature at which significant pair production can take place.

We then fit the broad-band spectra for the three epochs with the thermal Comptonization model, but now we allow $kT_{\rm e}$ of the two Comptonization clouds to be different. For the case of the average spectrum, we also include the PICSiT data. This improves the agreement with the data, but the fits remain rather poor. In particular, we obtain $\chi^2_\nu\approx 984/794$, 1835/1309, 2437/2013 for epochs 1, 2, A, respectively, with significant residuals at the highest energies. These models also require very strong reflection from the harder component, e.g., ${\cal R}_{\rm h}\approx 2.8$, 3 (constrained to $\leq$3) for epochs 2 and A, respectively, which puts in question their physical reality.

We then allow for the presence of a non-thermal tail in the electron distribution of the hard component, which adds two free parameters. We find that this strongly improves the fits, with the spectra and data/model ratios shown in Figure \ref{eeuf_ratio} and the parameters given in Table \ref{t_fits}. With the new values of $\chi^2$, the probabilities for the improvement to be by chance estimated by the F-test for epochs 1, 2, A equal $\approx 3\times 10^{-6}$, $3\times 10^{-14}$, $10^{-34}$ respectively. We find $\gamma_{\rm min}\approx 1.2$--1.3 and $p\approx 3.6$--5.6. The index for the average spectrum is significantly steeper than in the case of the individual observations. The cause of this difference remains unclear. The {\textit{INTEGRAL}}\xspace and {\textit{NuSTAR}}\xspace observations do not cover exactly the same periods and have different durations. It is possible that during some of the {\textit{INTEGRAL}}\xspace observations the high-energy tail was weaker than in the case of epochs 1--2. On the other hand, the fit to the average spectrum shows a visible hint of a hardening at $\gtrsim$0.5\,MeV. While the contribution to $\chi^2$ is only 10.2 for 13 bins, all of the best-fit values lie above the model, which is unlikely to happen by chance.

Apart from the presence of the non-thermal tail, the results shown in Table \ref{t_fits} are very similar to those for the {\textit{NuSTAR}}\xspace observations 2 and 4 analyzed by {\citetalias{Zdziarski21b}}\xspace, with the Fe abundance close to solar and the disk being substantially truncated. Since the average {\textit{NuSTAR}}\xspace spectrum is from 7 individual observations, some of fitted quantities may be biased by the averaging. In particular, we do not claim that the values of $R_{\rm in}$ and $Z_{\rm Fe}$ higher than those for epochs 1 and 2, and the very low $\Delta R$ are real. The value of the averaging is, rather, in its high statistical quality of the high-energy spectrum.

Instead of adding a non-thermal tail to the electron distribution, \citet{Roques19} was able to fit the SPI data alone by thermal Comptonization, reflection and an e-folded power law. However, the {\textit{NuSTAR}}\xspace data for this source require two reflection components \citep{Buisson19} as well as two incident continua with different shapes ({\citetalias{Zdziarski21b}}\xspace), which we have assumed in our joint fitting. We can still add another spectral component to our two-component thermal-Compton fit described above. We have found that an added e-folded power law has a very low spectral index, $\Gamma\sim 0$. In the absence of self-absorption (which is ruled out at high energies), a spectrum as hard can be obtained only from thermal Comptonization close to saturation, with a pronounced Wien peak. Also, an e-folded photon power law is cut off at high energies much faster than the synchrotron spectrum (usually invoked as a jet contribution) of power-law electrons with an exponential cutoff, which can be seen using eq.\ (9) of \citet{ZLG03}. We have thus replaced the e-folded power law by thermal Comptonization using the {\tt thcomp} model \citep{Z20_thcomp}, which, unlike {\tt compps}, is well suited for saturated Comptonization. For the avarage spectrum, we have obtained a reasonable fit with $\Gamma\approx 1.2$ and $kT_{\rm e}\approx 68$\,keV for this additional component. Still, this model yields $\chi^2_\nu\approx 2324/2010$, which is significantly higher than the model with a non-thermal tail (Table \ref{t_fits}), with $\Delta \chi^2\approx +69$, as well as the data are much above the model at $E\gtrsim 0.5$\,MeV. The two models are not nested, so the F-test cannot be used. Instead, the Akaike information criterion \citep{Akaike78} can be used, which yields the relative likelihood of the model with the additional component compared to that with the tail of $\approx\! 4\times 10^{-16}$, where we used it as given by eq.\ (1) in \citet{Dzielak19}. Thus, we see that an addition of a high-energy tail to the electron distribution yields a much better fit than adding an additional spectral component.

\section{Pair production}
\label{pair}

We express the photon-photon pair production rate through the differential photon density at its threshold, $m_{\rm e}c^2$, and express the photon energies as $\epsilon\equiv E/(m_{\rm e}c^2)$. We denote that density, ${\rm d}n/{\rm d}\epsilon$, at $\epsilon=1$ as $n_1$. In order to estimate it, we assume a simple model of a sphere with a radius, $R$, a uniform photon density, and the average photon escape time of $\approx \!(R/c) (1+g_{\rm C}\tau_{\rm T}/3)$, where $\tau_{\rm T}\equiv R n_{\rm e}\sigma_{\rm T}$, $g_{\rm C}\equiv \sigma_{\rm C}/\sigma_{\rm T}$, $\sigma_{\rm T}$ and $\sigma_{\rm C}$ are the Thomson and Compton cross sections, respectively, $n_{\rm e}$ is the total density of electrons and positrons, and $g_{\rm C}(\epsilon=1)\approx 0.43$. This yields
\begin{equation}
n_1\approx F_E(511{\rm keV})\frac{3(1+g_{\rm C}\tau_{\rm T}/3)d^2}{R^2 c},
\label{n1}
\end{equation}
where $F_E$ is in units of keV\,cm$^{-2}$\,s$^{-1}$\,keV$^{-1}$. Given that the spectra in the vicinity of 511\,keV can be well approximated by power laws, see Figures \ref{eeuf_ratio}, \ref{power_average}, we assume ${\rm d}n/{\rm d}\epsilon= n_1 \epsilon^{-\Gamma}$. We further approximate the photon field within the source as isotropic, allowing us to use the absorption coefficient for photon-photon pair production by \citet{GS67}\footnote{Corrected in \citet{Brown73}. See also the corrected first three coefficients of the series expansion in \citet{Zdziarski88}; the fourth is in error.}. The absorption coefficient for photons crossing a power-law isotropic photon field without cutoffs is given by eqs.\ (B3) and (B5) of \citet{Svensson87}, which corresponds to the radial optical depth of
\begin{equation}
\tau_{\gamma\gamma}(\epsilon)\approx \frac{7\epsilon^{\Gamma-1} n_1\sigma_{\rm T} R}{6\Gamma^{5/3} (1+\Gamma)}.
\label{taugg}
\end{equation}
In the absence of cutoffs, the total pair production rate is infinite. Here, we integrate the absorption coefficient corresponding to Equation (\ref{taugg}) over a power law with a high-energy cutoff at $\epsilon_{\rm c}\equiv E_{\rm c}/(m_{\rm e}c^2)>1$, and include a factor of 1/2 to account for double counting of pair-producing photons. Note that this neglects the effect of pair absorption, which we will consider later. This yields,
\begin{equation}
\dot n_{\gamma\gamma}\approx \frac{7 C \ln\epsilon_{\rm c}}
{6\Gamma^{5/3} (1+\Gamma)} n_1^2 \sigma_{\rm T} c.
\label{pprate}
\end{equation}
It depends logarithmically on the unknown high-energy cutoff. Equation (\ref{pprate}) also requires that the power law extends at low energies to at least $\epsilon=\epsilon_{\rm c}^{-1}$. The neglect of the cutoff in calculating the absorption coefficient means that the rate calculated as above is overestimated, which we account for by the factor $C<1$. At $\epsilon_{\rm c}=4$ and $\Gamma=3$, 4, $C\approx 0.71$, 0.78, respectively. Given the lack of an observed high-energy cutoff, the actual value of $\epsilon_{\rm c}$ remains unknown, but the average spectrum shown in Figure \ref{power_average} implies $\epsilon_{\rm c}\gtrsim 4$.

We assume the produced pairs lose quickly their energy in excess of the thermal one in Compton scattering and thermalize. The pair annihilation rate by thermal e$^\pm$ is
\begin{equation}
(\dot n_+)_{\rm ann} =\frac{3}{32} u(2-u) n_{\rm e}^2 \sigma_{\rm T}c g_{\rm A}(\theta),
\label{annrate}
\end{equation}
where $u\equiv 2 n_+/n_{\rm e}$ is the pair abundance, $n_+$ is the positron density, $\theta\equiv kT_{\rm e}/m_{\rm e}c^2$ and $g_{\rm A}$ is the correction to the Born rate, which can be fitted as \citep{Svensson82a}
\begin{equation}
g_{\rm A}\approx \left[1+2\theta^2/\ln(2\eta_{\rm E}\theta +1.3)\right]^{-1},
\label{gA}
\end{equation}
where $\eta_{\rm E}\equiv \exp(-\gamma_{\rm E})\approx 0.5616$ and $\gamma_{\rm E}$ is Euler's constant. Here, we neglect the small effect of annihilation of e$^\pm$ from the non-thermal tail of the fitted distribution, whose tail contains only a small fraction of all e$^\pm$.

We then consider escape of pairs, which is very likely to be present as advection to both the BH and the jet or winds. We parametrize it by the ratio of the light travel time across the source to the time scale of pair escape, $\beta_{\rm esc}\equiv (R/c)/[n_+/(\dot n_+)_{\rm esc}]<1$ (following eq.\ 18 of \citealt{Zdziarski85}). This gives
\begin{equation}
(\dot n_+)_{\rm esc}=\frac{1}{2\tau_{\rm T}}\beta_{\rm esc} u n_{\rm e}^2\sigma_{\rm T}c.
\label{escape}
\end{equation}
The rates of annihilation and escape become equal for
\begin{equation}
\beta_{\rm esc}=\frac{3}{16}(2-u)g_{\rm A}(\theta) \tau_{\rm T}
\label{beta_eq}
\end{equation}
(not assumed below). Pair equilibrium corresponds to $\dot n_{\gamma\gamma}= (\dot n_+)_{\rm ann}+(\dot n_+)_{\rm esc}$. Its solution gives the radius of
\begin{equation}
R_{\rm eq}\approx \frac{4(7 C \ln \epsilon_{\rm c})^{\frac{1}{2}}\sigma_{\rm T} d^2 (1+0.14\tau_{\rm T}) F_E(511{\rm keV})}{\Gamma^{5/6}[u(\Gamma+1)]^{\frac{1}{2}}\tau_{\rm T}c \left[(2-u)g_{\rm A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm T}}\right]^{\frac{1}{2}}},
\label{radius}
\end{equation}
which is $\appropto u^{-1/2}$. The minimum radius at which pair equilibrium can be established, which we denote as $R_{\rm pair}$, corresponds to the pair dominance, i.e., $u\rightarrow 1$. Also, for a known radius, e.g., setting $R_{\rm eq}=R_{\rm in}$ of Table \ref{t_fits}, we can solve for $u$. For a given $u$, the optical depth to pair production can be calculated using Equation (\ref{taugg}),
\begin{equation}
\tau_{\gamma\gamma}(\epsilon)\approx \frac{\epsilon^{\Gamma-1}\tau_{\rm T} (7 u)^{\frac{1}{2}} \left[(2-u)g_{\rm A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm T}}\right]^{\frac{1}{2}} } {8\Gamma^{5/6}(\Gamma+1)^{\frac{1}{2}}(C \ln \epsilon_{\rm c})^{\frac{1}{2}} },
\label{taugg1}
\end{equation}
which is maximized at $u=1$, or, equivalently, $R=R_{\rm pair}$.

\renewcommand{\arraystretch}{0.9}
\setlength{\tabcolsep}{2pt}
\begin{table}\centering
\caption{High-energy and e$^\pm$ pair parameters
}
\vskip -0.4cm
\begin{tabular}{lcccccc}
\hline
Epoch &$\Gamma$ &$\frac{E F_E(511{\rm keV})}{{\rm keV\,cm}^{-2}\,{\rm s}^{-1}}$ &$\tau_{\rm T}$ &$\frac{R_{\rm pair}}{R_{\rm g}}$ &$\left.\tau_{\gamma\gamma}\right(u\!=\!1)$ &$u(R_{\rm in})$\\
\hline
1 & $3.60^{+0.26}_{-0.24}$ & $2.48_{-0.54}^{+0.62}$ &6.1 & 3.9 & 0.32& 0.018\\
2 & $3.70^{+0.22}_{-0.20}$ & $2.35_{-0.43}^{+0.48}$ &5.1 & 3.7 & 0.29& 0.011\\
A & $3.75^{+0.05}_{-0.04}$ & $2.09_{-0.09}^{+0.09}$ &5.5 & 3.0 & 0.34& 0.0005\\
\hline
\end{tabular}
\tablecomments{
$\Gamma$ and $E F_E(511{\rm keV})$ are for the fits at $E\gtrsim$160\,keV. The Thomson depth of the hard component, $\tau_{\rm T}$, the radius for $u=1$, $R_{\rm pair}$, the photon-photon optical depth at $u=\epsilon=1$, $\tau_{\gamma\gamma}$, and $u$, the pair abundance for $R_{\rm eq}=R_{\rm in}$, are given for the best-fit parameters, and $M=8{\rm M}_{\sun}$, $\epsilon_{\rm c}=4$, $\beta_{\rm esc}=0.3$, $C=0.75$.}
\label{pairs}
\end{table}

\begin{figure}
\centering
\includegraphics[width=6.cm]{isgri_picsit_spi_average_power-eps-converted-to.pdf}
  \caption{The average data above 160\,keV (normalized to the SPI) fitted with a power law: SPI (blue), ISGRI (red), PICsIT (black). Including an e$^\pm$ pair annihilation line in the fit yields its null normalization. We plot the maximum line allowed by the data at $2\sigma$.
}\label{power_average}
\end{figure}

We then calculate the total observed flux of photons from pair annihilation as $\dot N_{\rm ann}= 2(\dot n_+)_{\rm ann}(4\pi R^3/3)/[4\pi d^2(1+0.14\tau_{\rm T})]$ (in units of cm$^{-2}$\,s$^{-1}$), where the factor of 2 accounts for two photons produced in each annihilation event and the flux is reduced by scattering (and we neglect secondary pair production by the annihilation photons). Using $R=R_{\rm eq}$, we find
\begin{equation}
\dot N_{\rm ann}\approx \frac{(7 C u\ln \epsilon_{\rm c})^{\frac{1}{2}}(2-u)\tau_{\rm T} F_E(511{\rm keV})}{4\Gamma^{5/6} (\Gamma+1)^{\frac{1}{2}}\left[(2-u)g_{\rm A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm T}}\right]^{\frac{1}{2}}}.
\label{obs_ann}
\end{equation}
The above rate assumes that annihilation of pairs escaping the hot plasma is negligible. In the limit of $kT_{\rm e}\ll m_{\rm e}c^2$ (which is satisfied in {MAXI J1820+070}\xspace), the observed energy spectrum ($F_{E,{\rm ann}}$ in units of keV\,cm$^{-2}$\,s$^{-1}$\,keV$^{-1}$) from pair annihilation becomes \citep{SLP96}
\begin{equation}
F_{E,{\rm ann}}\approx \dot N_{\rm ann}\frac{\epsilon^{5/2}}{(\pi \theta)^{1/2}}\exp\left[-\frac{(\epsilon-1)^2}{\epsilon\theta}\right].
\label{anspec}
\end{equation}
Here, we assumed that the annihilating electrons are purely thermal. The steady-state electron distribution for epoch A contains about 10\% of non-thermal electrons, which would result in a weak high-energy tail to the above spectrum.

We now use the data from the {MAXI J1820+070}\xspace observations to determine the values of the parameters related to pair production. We have fitted the spectra of the three epochs at $E\gtrsim 160$\,keV by power laws, and show the results in Table \ref{pairs} and in Figure \ref{power_average} for the average spectrum. Then we use equation (\ref{radius}) assuming $\epsilon_{\rm c}=4$, $C=0.75$ and $\beta_{\rm esc}=0.3$, and obtain the radii for the pair dominance as given in Table \ref{pairs}, expressed in units of the gravitational radius for the assumed BH mass of $8{\rm M}_{\sun}$. We find $R_{\rm pair}\approx 3$--4$R_{\rm g}$ and, from equation (\ref{taugg1}), the corresponding $\tau_{\gamma\gamma} (\epsilon=1) \approx 0.3$. At such low radii, GR effects would increase the photon energies and density, and lead to some increase of $R_{\rm pair}$. On the other hand, attenuation of the power law at $\epsilon\gtrsim 1$ would decrease $R_{\rm pair}$. However, the obtained values of $R_{\rm pair}$ are smaller by an order of magnitude than the fitted disk inner radii, $R_{\rm in}$, given in Table \ref{t_fits}, below which the disk is replaced by a hot, pair-producing, flow. By solving $R_{\rm eq}=R_{\rm in}$, we find $u\ll 1$, Table \ref{pairs}. Thus, the hot plasma within $R_{\rm in}$ seems not to be dominated by pairs. In this case, $\tau_{\gamma\gamma}\ll 1$, and our neglect of pair absorption is justified. We also find the photon-particle pair production rate is negligible in this source.

The lack of a noticeable pair density is confirmed by including the annihilation line in the fit to the average spectrum, assuming the best-fit value $kT_{\rm e,h}=16.4$\,keV. We have found that the best-fit normalization of the line is null, which also does not allow us to constrain the presence of a non-thermal tail of the annihilation spectrum. We show the annihilation spectrum corresponding to the $2\sigma$ upper limit in Figure \ref{power_average}. This limit corresponds to $\lesssim$0.130 of the normalization corresponding to the pair-dominated plasma ($u=1$), which in turn corresponds to $u\lesssim 0.0075$ and $R_{\rm eq}\gtrsim 27 R_{\rm g}$.

\section{Discussion}
\label{discussion}

In the hard state, we have obtained the disk inner truncation radii of $R_{\rm in}\approx 20$--$30 R_{\rm g}$ (for spectral fitting to the individual observations), while the equilibrium radii corresponding of pair dominance are $R_{\rm pair}\sim 3$--$4 R_{\rm g}$. This then implies a low pair abundance in the hot, hybrid, flow downstream of $R_{\rm in}$, which produces the observed high-energy spectra. Still, $R_{\rm pair}\sim 3$--$4 R_{\rm g}$ hints at the importance of pair production in regulating the physical state of the flow. If pairs were just unimportant, the minimum radius limited by pair equilibrium could have any value, but instead we find $R_{\rm pair}\sim R_{\rm ISCO}$, suggesting that pair production somehow knows the fundamental size scale of the system. The pairs produced by the non-thermal photon tail may contribute to regulation of the temperature of the thermal component of the electron distribution \citep{Coppi99,Fabian17}.

Alternatively, the obtained low values of the electron temperature, $kT_{\rm e}\sim 20$\,keV, could be due to the energy balance in the flow itself.  \citet{PV09} and \citet{MB09} suggested $T_{\rm e}$ can be regulated by thermalization in a synchrotron boiler \citep{GGS88}. Such regulation also takes place in advection-dominated accretion flows (ADAFs), but the electron temperatures of such flows are much higher, e.g., \citet{YN14}. On the other hand, the existing ADAF models consider only thermal electrons, while the presence of non-thermal tails can greatly increase the synchrotron cooling (see below), which effect, could, in principle, reconcile this model with the data. Some studies of extended accretion flows with hybrid electrons were performed by \citet{VPV13}, but without including heating from ions.

We note that the {\textit{NuSTAR}}\xspace data were fitted by \citet{Buisson19} with $R_{\rm in}\approx 5 R_{\rm g}$ and two sources on each side of the BH on its rotation axis, with the lower one at $\approx 3$--$5 R_{\rm g}$, and at $i\approx 30$--$35\degr$. This spectral solution would imply pair-dominated plasmas. As discussed in {\citetalias{Zdziarski21b}}\xspace, the {\textit{NuSTAR}}\xspace data for this source yield two separate solutions, one low-$i$ and low-$R_{\rm in}$, and one high-$i$, high-$R_{\rm in}$. However, the low-$R_{\rm in}$ is ruled out by the binary and jet measurements, showing $i\geq 60\degr$ \citep{Torres20,Atri20}, leaving the high-$R_{\rm in}$ solution as applicable to this source.

An important diagnostic, confirming the low pair dominance independently of the pair equilibrium calculations, is the absence of an annihilation feature in the observed spectrum. Purely thermal plasmas do not show distinct pair annihilation lines even if dominated by pairs \citep{Z86,Stern95,Svensson96}. On the other hand, hybrid plasmas can emit X-ray spectra well reproduced by Comptonization on mostly thermal electrons, but can still exhibit relatively narrow annihilation features. The absence of such a feature in the average spectrum puts therefore strong constraints on the pair abundance.

We should also point out that the {\tt compps} code assumes the high-energy tail has the form of $\gamma^{-p}$, while physical acceleration processes lead to a power law in either kinetic energy or momentum. Given the relatively low values of $\gamma_{\rm min}\approx$1.2--1.3 we have obtained from the fitting, the electron distribution consists of a Maxwellian part, and then a relatively flat part steepening only at $\gamma\gtrsim 3$ or so. It is likely that our quantitative results on the electron distribution would be affected by this assumption, though not our overall conclusions.

With this caveat, the fitted steady-state power-law indices of the electron tail are in the range of $p\approx 3.6$--5.6. Then, the power-law fits to data above 160 keV give even steeper indices, with $p=2\Gamma-1\approx 6.2$--6.5 implied by $\Gamma$ given in Table \ref{pairs}. The Compton (in the Thomson limit) and synchrotron energy losses steepen the distribution of accelerated electrons by unity, giving their index as $p_{\rm acc}=p-1$. Except for the case of the hybrid fit to the broad-band data for epoch 1, the implied values of $p_{\rm acc}$ are steeper than those from shock acceleration. In particular, $p_{\rm acc}\approx 2.3$ for collisionless shocks in pair plasma \citep{Sironi15}, though the presence of ions steepens that \citep{Ball18}. On the other hand, magnetic reconnection can readily give particle spectra of similar indices \citep{Sironi16,Sironi20}.

We can compare our results with those for the accreting BH binary Cyg X-1. Its high-energy spectra were fitted by \citet{McConnell02} also using {\tt compps}. In the hard state, they obtained $kT_{\rm e}= 58^{+18}_{-22}$\,keV, $\gamma_{\rm min}=1.39^{+0.51}_{-0.34}$, $p=5.4^{+0.4}_{-0.3}$. The electron temperature in that case is substantially higher than in {MAXI J1820+070}\xspace, but the tail parameters are similar.

The presence of high-energy tails in the electron distribution likely results in the synchrotron emission of the electrons being greatly enhanced with respect to the Maxwellian case \citep{WZ01,VVP11,VPV13, PV14}. This is likely to result in the synchrotron emission becoming an important source of the seed photons for Comptonization, with a typical self-absorption turnover at $\lesssim$10\,eV. We have thus run spectral models in which the seed photon temperature was set to 10\,eV, and found similar values of $\chi^2$ and minor differences in the best-fit parameters with respect to our main calculations (with $kT_{\rm bb}=200$\,eV). Thus, both the disk blackbody and synchrotron emission can be the dominant seed photons, with the spectral fits at $>$3\,keV not allowing us to distinguish between these possibilities.

\section{Conclusions}

We have studied two joint spectra from the 2018 hard-state rise of {MAXI J1820+070}\xspace from {\textit{NuSTAR}}\xspace and {\textit{INTEGRAL}}\xspace. We have found an excellent agreement between the {\textit{NuSTAR}}\xspace and SPI spectra in the overlapping region of 23--78\,keV, and an overall agreement between {\textit{NuSTAR}}\xspace and ISGRI. We have found the resulting 3--650\,keV spectra are well fitted by a two-component Comptonization model, with each component having different spectral index and separate reflection regions. However, the harder component, dominant at high energies, is emitted by hybrid electrons, whose distribution consists of a Maxwellian and a high-energy tail. The presence of the tail is required at a very high significance.

Except for the presence of the tail, the parameters fitted to the spectra are similar to those fitted by {\citetalias{Zdziarski21b}}\xspace to the {\textit{NuSTAR}}\xspace data of observations taken within the same state of the source (in particular those of their epochs 2 and 4). In particular, we find the disk is truncated at $\approx$20--$30 R_{\rm g}$. The electron temperatures are low, $\approx$20\,keV for the harder component, even lower than $\approx$30\,keV found in {\citetalias{Zdziarski21b}}\xspace. The optical depth is relatively large, $\tau_{\rm T}\approx 5$--6.

We have also obtained an average hard-state spectrum of {MAXI J1820+070}\xspace, based on the data spanning 53 d. This spectrum is quite similar to those of the two individual observations, but extends to $\approx$2\,MeV. It also includes the PICSiT data, which are found to be in an excellent agreement with those of the SPI. The high-energy tail in the average spectrum is steep, $\Gamma\approx 3.7$--3.8 at energies $>$160\,keV. The broad-band fit with the hybrid electrons yields the index of the tail of $p\approx 5.5$--5.6. The steepness of the tail favors magnetic reconnection as the acceleration process.

The average spectrum crosses 511\,keV, the threshold for e$^\pm$ pair production. We have developed a formalism calculating the rates of pair production, annihilation and escape in hybrid plasmas, and obtained formulae for the equilibrium radius at a given pair abundance and the optical depth to pair absorption. We have found that the minimum possible radius from the equilibrium condition, at which the plasma is pair-dominated is $\approx\! 3$--$4 R_{\rm g}$. This is, however, much less than our estimates of the size of the hot plasma, $R\sim R_{\rm in}\sim 20$--$30 R_{\rm g}$. At such radii, the equilibrium pair abundance is $\ll$1. This conclusion is confirmed by the absence of an annihilation feature in the average spectrum. Even at its $2\sigma$ upper limit, the implied size of the hot plasma is $\gtrsim 30 R_{\rm g}$, corresponding to a low pair abundance.

We consider it possible that pair production still acts as a thermostat, limiting the electron temperature. On the other hand, the low temperature can be due to an efficient cooling by synchrotron photons, which are efficiently emitted by hybrid plasmas.

\section*{Acknowledgments}
This work is based on observations with {\textit{INTEGRAL}}\xspace, an ESA project with instruments and Science Data Center funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Switzerland, Spain) and with the participation of Russia and the USA. The {\textit{INTEGRAL}}\xspace SPI project has been completed under the responsibility and leadership of CNES. The SPI team is grateful to ASI, CEA, CNES, DLR, ESA, INTA, NASA, and OSTC for their support. We acknowledge support from the International Space Science Institute (Bern), and the Polish National Science Centre under the grants 2015/18/A/ST9/00746, 2019/35/B/ST9/03944 and 2014/13/B/ST9/00570. AV and JP acknowledge the Academy of Finland grants 309308 and 333112, respectively.