detections a novelty, and some of them laying close to the sensitivity
detection threshold of the instrument, we describe in detail the VHE data
analysis, the calculation of the significance excess at the GRB position
(following the usual prescription used for VHE sources presented in Li and Ma
(1983)), and the methods adopted for the derivation of the spectral energy
distribution (SED) and of the light-curves. For each GRB, we also present the
interpretations that have been put forward in the literature. A discussion on
the main common properties and differences among this initial population of
VHE GRBs and with respect to the whole GRB population is presented in section
5, where we address also the question of what we have been learning from these
few detections.

In this section all quoted times refer to the time elapsed from the trigger
time $T_{0}$ of the Swift-BAT or Fermi-GBM instrument, as will be specified.
Photon indices are given in the notation $N_{\nu}\propto\nu^{\alpha}$, while
temporal indices are defined by $F(t)\propto t^{\beta}_{T}$.

###  4.1 GRB 160821B

GRB 160821B is a short GRB at $z=0.162$ detected by the Swift-BAT Siegel et
al. (2016) on 21 August 2016 at $T_{0}=22:29:13$ UT and by the Fermi-GBM
Stanbro and Meegan (2016). The analysis of MAGIC observations shows a $\sim
3\sigma$ excess at the GRB position.

####  4.1.1 General properties and multi-wavelength observations

The BAT prompt spectrum ($T_{90}=0.48$ s) is well described by a power-law
with index $\alpha=-0.11\pm 0.88$ and an exponential high-energy cutoff,
corresponding to a peak energy with $E_{p}=46.3\pm 6.4$ keV Palmer et al.
(2016). The GBM prompt spectrum ($T_{90}=1.088\pm 0.977$ s) is fitted with a
cutoff power-law function as well, with $E_{p}=92\pm 28$ keV. Being located at
redshift $z=0.162$, this is one of the nearest short GRBs detected up to date.
Its isotropic-equivalent energy $E_{\gamma,iso}\sim 1.2\times 10^{49}$ erg is
toward the low energy edge of the known distribution for short GRBs Berger
(2014)).

Afterglow observations are available in the radio, optical, X-ray and (V)HE
band. Fermi-LAT observations were performed from the trigger time up to $2315$
s and from $5285$ s to $8050$ s and did not reveal any significant excess in
the 0.3-3 GeV band Acciari et al. (2021). Swift-XRT observations Evans et al.
(2016) started $57$ s after the trigger time and revealed a complex behaviour
of the X-ray afterglow light curve. An initial plateau is followed by a steep
decay at around $10^{2}$ s. Then, a power-law decay with index $\sim-0.8$ is
observed after $\sim 10^{3}$ s Sbarufatti et al. (2016); Troja et al. (2019).
Optical observations were performed by several instruments Jeong et al.
(2016); Troja et al. (2016); Xu et al. (2016); Levan et al. (2016) revealing
the presence of a fading source with a magnitude $r=22.6\pm 0.1$ mag $0.95$
hrs after $T_{0}$. The identification of the host galaxy allowed the
measurement of the spectroscopic redshift $z=0.162$. The GRB is located in the
outskirts of the host spiral galaxy, at $\sim 15$ kpc projected distance from
its center Troja et al. (2019); Lamb et al. (2019). A fading radio afterglow
counterpart was observed at 6 GHz by VLA starting from 3.6 h after the burst
trigger Fong et al. (2016). The multi-wavelength light-curves of GRB 160821B
are shown in Figure 11.

####  4.1.2 VHE observations and results

The MAGIC telescopes started the follow-up of GRB 160821B with a very short
delay of $24$ s after $T_{0}$ and continued observations for about $4$ h
Acciari et al. (2021). The observations were performed with a relatively high
Night Sky Background (NSB) (2-8 times brighter than in dark nights) due to the
presence of the Moon, and in mid-high zenithal angle conditions (from 34∘ to
$55^{\circ}$). Unfortunately, the first $\sim$ 1.7 h of the data were strongly
affected by clouds. As a result, dedicated and optimized software
configurations were used. A more stringent image cleaning with respect to dark
conditions was applied to take into account the spurious contribution coming
from the high NSB. The analysis required cuts optimized on the Crab Nebula and
on Mrk421 observed in similar conditions, and correction factors for the low
atmospheric transmission, calculated thanks to the LIDAR facility Fruck, C. et
al. (2014).

Figure 10: GRB 160821B: angular distance distribution $\theta^{2}$ between the
nominal source position and the reconstructed event arrival directions. The
gray histogram represents the background events, while the black points with
blue crosses are the $\gamma$-like events. The vertical dashed line marks the
$\theta^{2}$ cut value and defines the region in which excess events and
signal significance are calculated. From Acciari et al. (2021). ©AAS.
Reproduced with permission.

The pre-trial analysis showed the presence of a $3.1\sigma$ ($2.9\sigma$ post-
trialA trial factor of 2 is considered due to the two sets of analysis cuts
used for MAGIC data analysis.) significance excess at the GRB position
provided by Swift-XRT (see Figure 10). The flux has been estimated for
energies above $0.5$ TeV assuming a power-law spectrum with photon index
$\alpha=-2$. In the first 1.7 h, where data taking was affected by bad
atmospheric transmission, only flux upper limits could be derived. This time
window has been divided into two time intervals ($24-1216$ s and $1258-6098$
s) and the derived upper limits are respectively $1.1\times 10^{-11}$ cm-2s-1
and $5.4\times 10^{-12}$ cm-2 s-1. In the subsequent 2.2 h ($6134-14130$ s)
assuming that the signal is real, a flux value could be calculated and is
$9.9\pm 4.8\times 10^{-13}$ cm-2 s-1. For the same time interval, also a flux
upper limit has been estimated and gives $3.0\times 10^{-12}$ cm-2 s-1. All
the mentioned fluxes are shown in Figure 11 (red symbols) and refer to the
observed values, i.e., without correcting for EBL absorption. Upper limits
have been calculated at 95$\%$ confidence level following the prescriptions of
Rolke, W.A. et al. (2005). The low ($\sim 3\sigma$) significance estimated did
not allow to obtain an unfolded spectrum. As a result, in the SED (Figure 12)
the reconstructed flux in the third bin ($6134-14130$ s) over the energy range
$0.5-5$ TeV is represented as an error box. The statistical error on the
photon flux has been taken into account, while, for simplicity, the systematic
error for the assumed spectral index was neglected. The flux inferred by MAGIC
observations in the $0.5-5$ TeV energy range would imply a TeV luminosity at
least 5 times larger (when de-absorbed by EBL) than the luminosity emitted in
the X-ray band.

####  4.1.3 Interpretation

A modeling of the multi-wavelength observations, including MAGIC data, has
been presented by the MAGIC Collaboration in Acciari et al. (2021). The
emission is interpreted as the sum of several components, dominating at
different times and in different energy bands:

  * •

synchrotron and SSC emission from electrons accelerated at the forward shock;
this is in general the dominant emission component;

  * •

synchrotron emission from electrons accelerated by the reverse shock, which is
found to dominate the radio band until $t\sim 4.8$ h;

  * •

kilonova emission, powered by freshly synthesized $r$-process elements
released in neutron star mergers; this component is found to dominate the
optical/nIR from around 1 to 4 days Troja et al. (2019); Lamb et al. (2019);

  * •

an X-ray extended emission component, widely attributed to long-lasting
activity of the central engine, here dominating the X-ray band for $t<10^{3}$
s.

In performing this multi-component modeling, the synchrotron and SSC forward
shock emission have been calculated with a one-zone numerical code (see et al.
(2019) for details), while the reverse shock and kilonova emission
contributions have been taken from Troja et al. (2019). Only X-ray data at
$t>10^{3}$ s have been included in the modeling, to exclude the extended
emission component. The broad-band modeling is shown over-plotted to the
light-curves in Figure 11 (solid lines) and to the SED between 1.7 and 4 h in
Figure 12.

Figure 11: GRB 160821B: multi-wavelength light curves (from radio to TeV) and
their modeling according to Acciari et al. (2021). ©AAS. Reproduced with
permission. The different contributions from the forward shock (FS), reverse
shock (RS), and kilonova are shown (see legend). Figure 12: GRB 160821B:
modeling of the simultaneous multi-wavelength SED at approximately $\sim 3$ h
according to Acciari et al. (2021) (©AAS. Reproduced with permission.), for
the same parameters used to model the lightcurves in Figure 11. The shaded
areas show the sensitivity energy range of the different instruments. The
MAGIC error box on the reconstructed flux is also shown. Synchrotron (solid
black line), intrinsic SSC (before EBL absorption, dashed black line) and SSC
emission with EBL attenuation (solid red line) estimated from the numerical
modeling are shown.

A very good agreement between data and modeling is found in radio (green lines
and points), optical (yellow and pink lines and points) and X-rays (blue lines
and points). A large degeneracy is present in the parameters, and the data
modeling only allows to identify the ranges for the permitted values of each
parameter. These are reported in Table 1 and we note that they are very
similar to those estimated in Troja et al. (2019) and in a later work by Zhang
et al. (2021). In the allowed parameter space defined by radio, optical and
X-ray observations, different combinations of the parameters predicts
different SSC fluxes at 1 TeV are found, reaching at most
$F_{SSC}^{(1TeV)}\sim 2\times 10^{-13}$ erg cm-2 s-1. This value, when
attenuated by EBL, is at least one order of magnitude fainter than the one
inferred from data analysis of the MAGIC observations. In other words, the
parameter space constrained by the observations at lower frequencies is unable
to account for such energetic TeV emission, and the SSC forward shock scenario
fails to reproduce the observations, provided that the hint of excess found by
MAGIC is a real signal from the source.

An alternative scenario that has been explored is the external inverse Compton
(EIC) scenario, investigated by Zhang et al. (2021). These authors first
consider a one-zone SSC model, and reach similar conclusions to those
presented by the MAGIC Collaboration Acciari et al. (2021): the SSC mechanism
predicts a TeV flux around 1-2 orders of magnitude lower than the MAGIC
observations (see Figure 13, orange curves). The alternative EIC scenario is
then considered by the authors, where the seed photons are provided by the
extended X-ray emission and the X-ray plateau. The extended emission and the
plateau are fitted using two phenomenological functions. The energy spectrum
of the late-prompt emission is described by a broken power-law (see Figure 13,
top and bottom panels). For the EIC model, the VHE spectrum is inferred for
three different observed times ($t=1.1,1.8,2$ h) and compared to the MAGIC
flux averaged between 1.7 and 4 h. As it can be seen in Figure 13 (bottom
panel), the model flux at 2 h under-predicts the MAGIC flux (the MAGIC
observed flux, green shaded area, should be compared with the EBL-absorbed
model flux). We conclude that also the EIC model is unable to explain the
large TeV flux suggested by MAGIC observations.

Figure 13: GRB 160821B: modeling of multi-wavelength light curves (top panel)
and simultaneous SED (bottom panel) in radio, optical, X-ray and VHE band
presented by Zhang et al. (2021).©AAS. Reproduced with permission. Two
scenarios are considered: SSC (dashed orange lines) and EIC (dash-dotted green
lines). In the top panel the flux contribution in the different energy bands
due to the forward shock (FS) both in the SSC and in the EIC scenario, the
extended X-ray emission (EE) and the X-ray plateau emission (PL) are shown. In
the bottom panel for the VHE band both the intrinsic (thin lines) and the
absorbed (thick lines) VHE curves are shown for the SSC and EIC scenarios for
three different observed times. For the lower energies the broken power-law of
the late prompt emission (dashed violet line) and the expected synchrotron
SEDs (solid red line) are displayed. The MAGIC observations (green symbols and
green shaded area) are absorbed by EBL.

$E_{k}$ log($\epsilon_{e}$) log($\epsilon_{B}$) log(n) $p$ $\xi_{e}$
$\theta_{j}$ erg cm-3 rad MAGIC Coll. $10^{51}-10^{52}$ [-1 ; -0.1] [-5.5 ;
-0.8] [-4.85 ; -0.24] 2.2-2.35 1 / Troja + 2019 $10^{50}-10^{51}$ [-0.39 ;
-0.05] [-3.1 ; -1.1] [-4.2 ; -1.7] 2.26-2.39 1 0.08-0.50 Zhang + 2021 (SSC)
$3\times 10^{51}$ -0.52 -5 -1.3 2.3 0.5 0.15 Zhang + 2021 (EIC) $2\times
10^{51}$ -0.3 -6 -1 2.5 0.1 0.1

Table 1: GRB 160821B. List of the best fit parameters inferred from multi-
wavelength modeling of the afterglow radiation by different authors.

###  4.2 GRB 180720B

GRB 180720B is a long GRB at $z=0.654$ triggered on 20 July 2018 by the Fermi-
GBM Roberts and Meegan (2018) at $T_{0}=14:21:39$ UT and 5 s later by the
Swift-BAT instrument Siegel et al. (2018). The H.E.S.S. telescopes observed
and detected GRB 180720B about 11 h after the prompt emission, with a $\sim
5\sigma$ statistical significance.

####  4.2.1 General properties and multi-wavelength observations

The extremely bright prompt emission of this event, which is the seventh in
brightness among the GRBs detected by the Fermi-GBM until then, lasted for
$T_{90}=48.9\pm 0.4$ s and released an isotropic-equivalent energy
$E_{\gamma,iso}=6.0\pm 0.1\times 10^{53}$ erg in the 50-300 keV range.

Figure 14: GRB 180720B: broad-band light curves and photon index evolution
including optical, X-ray and VHE data. For the late time H.E.S.S. observation
(18 days after the trigger) an UL has been derived. A flux temporal decay with
index $1.2$ (similar to what observed in optical and X-ray) is also shown in
black dashed line. From Abdalla et al. (2019).

Multi-wavelength afterglow observations covered the entire electromagnetic
spectrum (see Figure 11). Significant signal was detected by Fermi-LAT from
the trigger time up to 700 s, with the highest photon energy of 5 GeV detected
137 s after the burst trigger Bissaldi and Racusin (2018). The Swift-XRT
telescope observed and identified a bright afterglow starting from $90$ s.
This was still visible almost 30 days after the trigger time. The late-time
light curve (from $2\times 10^{3}$ s to $4\times 10^{6}$ s) can be modelled
with an initial power-law decay with an index $-1.19^{+0.01}_{-0.02}$ followed
by a break at $t_{break}=8\times 10^{4}$ s to an index of
$-1.55^{+0.04}_{-0.05}$ https://www.swift.ac.uk/xrt_live_cat/00848890/.
Several optical observations Sasada et al. (2018); Martone et al. (2018); Reva
et al. (2018); Watson et al. (2018); Schmalz et al. (2018); Zheng and
Filippenko (2018); Lipunov et al. (2018); Itoh et al. (2018); Izzo et al.
(2018); Covino and Fugazza (2018) revealed the presence of a counterpart and
allowed to estimate the redshift value of $z=0.654$. The optical afterglow was
seen to be slowly fading at an almost constant rate from around 10-11 h after
the trigger time Jelinek et al. (2018); Kann et al. (2018) as discussed in
Rhodes et al. (2020). Radio observations (not shown in the figure) were also
performed starting from $\sim 1.7$ days after the burst showing a steep power-
law decaying emission Rhodes et al. (2020).

####  4.2.2 VHE observations and results

The H.E.S.S. telescopes followed-up the event for $\sim 2$ h starting from
10.1 h, revealing the presence of a source with a $5.3\sigma$ pre-trial
significance ($5.0\sigma$ post-trialpost-trial significance is estimated by
accounting for the previously well-localized GRBs observed by H.E.S.S. in the
same array configuration as of GRB 180720B). The observation was performed in
standard dark and good weather conditions with a mean zenith angle of 31.5∘.
Another observation was performed under similar conditions 18 days after the
previous one with results consistent with background events. The inferred flux
at $\sim 11$ h and the flux upper limit at 18 d are shown in Figure 14 (red
symbols).

The observed spectrum in the $0.1-0.44$ TeV energy band has been fitted both
with a power-law (panel on the left in Figure 15) and with a power-law with a
cutoff $F_{int}=F_{0,int}\big{(}\frac{E}{E_{0}}\big{)}^{-\gamma_{int}}$, to
describe an intrinsic power-law spectrum affected by the EBL attenuation
$e^{-\tau(E,z)}$ (see Figure 15, panel on the right). With reference to the
second fit, the analysis returns a photon index $\gamma_{int}=1.6\pm 1.2$
(statistical) $\pm 0.4$ (systematic) and a flux normalization
$F_{0,int}=(7.52\pm 2.03$ (statistical) ${}^{+4.53}_{-3.84})$ (systematic)
$\times 10^{-10}$ TeV-1 cm-2 s-1, evaluated at an energy $E_{0,int}=0.154$
TeV.

Figure 15: GRB 180720B: spectral fit to the observed emission in the
$0.1-0.44$ TeV energy range. In the left panel the observed spectrum is
assumed to be described by a power-law model, while in the panel on the right,
the spectrum is an power-law attenuated by EBL. Statistical and systematic
uncertainties at $1\sigma$ confidence level are shown with shaded areas. From
Abdalla et al. (2019).

####  4.2.3 Interpretation

The H.E.S.S. Collaboration explored two possible radiation mechanisms to
explain the VHE emission from GRB 180720B Abdalla et al. (2019): synchrotron
emission and SSC radiation. A full modeling is not performed, and the
discussion and comparison among the two different scenarios is based on
estimates of the typical and maximal electron energy necessary in the two
cases and on the comparison between spectral and temporal indices in different
energy ranges. A synchrotron spectrum with a flat ($\alpha\sim-2$) slope
extending from X-ray to VHE could model the emission with one single broad
component and explain the similarity between the H.E.S.S., Fermi-LAT, and
Swift-XRT luminosities, and the consistency among their photon index values.
The large error on the VHE photon index however is not placing strong
constraints, leaving open both the possibility of a consistency with the
extrapolation of the synchrotron spectrum but also the possibility of a
spectral hardening, indicative of a second component. A synchrotron origin of
$10^{2}$ GeV photons would require to find a process able to accelerate
electrons up to PeV energies, which is in excess of the maximum electron
energy achievable in external shocks (for a discussion, see Section 2.3.1).
Adopting the standard Bhom limit, $>100$ GeV emission 10 h after the burst
would require a huge bulk Lorentz factor $\Gamma\sim 1000$ which at these late
times is really unlikely. As a result, these strong requirements disfavour the
synchrotron emission as responsible of the VHE component in GRB 180720B.

Figure 16: GRB 180720B: modeling of the broad-band light curves (top panel)
and SED at the time of the H.E.S.S. detection (bottom panel) proposed by Wang
et al. (2019). Both the synchrotron and the SSC contribution to the total flux
are shown (see legend). In the SED, X-ray and H.E.S.S. data are shown
respectively with the green and the blue boxes.

The SSC scattering, on the contrary, arises as a natural candidate. A full
broad-band modeling of GRB 180720B data in this scenario is presented in Wang
et al. (2019). A numerical code reproducing the synchrotron and SSC emission
in the afterglow shocks has been used (see Liu et al. (2013)). The resulting
light-curves and SED in the H.E.S.S. observational time window are shown in
Figure 16. The full emission is explained as afterglow forward shock radiation
(except for the initial peak in the optical and X-ray curves at $t\sim 10^{2}$
s which is attributed to reverse shock emission). In the case of a constant-
density ISM environment, the parameters that best reproduce the data are:
$E_{k}=10^{54}$ erg, $n=0.1$ cm-3, $\epsilon_{e}=0.1$, $\epsilon_{B}=10^{-4}$,
$\Gamma_{0}=300$ and $p=2.4$. As it can be noticed, the equipartition factor
$\epsilon_{B}$ needs to assume a quite low value in order to explain the
observations. A stellar wind-like environment is discarded by the authors on
the basis of the comparison between the expected flux at $\sim 1-10$ GeV,
following the prescriptions of Sari, Re’em and Esin (2001) ($\gtrsim 2\times
10^{-6}$ erg cm-2 s-1 at $t\approx 100$ s) and the one observed by Fermi-LAT
($\sim 10^{-8}$ erg cm-2 s-1 at $t\approx 100$ s). A low magnetic field
equipartition factor is derived from the condition $E_{KN}\gtrsim 0.44$ TeV at
$t\approx 10$ h where $E_{KN}$ is the energy at which the KN scattering
becomes relevant. A transition energy between synchrotron and SSC component of
$\sim 1$ GeV is derived. Such value falls into the Fermi-LAT energy range and
is compatible with a hardening of the spectrum in the VHE band. However, since
Fermi-LAT sensitivity is above the predicted flux of GRB 180720B at $10$ h in
the GeV band, the data cannot firmly confirm the presence of this transition.

###  4.3 GRB 190114C

GRB 190114C is a long GRB at redshift $z=0.42$ triggered by the Swift-BAT
Gropp et al. (2019) on 14 January 2019 at $T_{0}=20:57:03$ UT, and by the
Fermi-GBM Hamburg et al. (2019). The event was detected also by several other
space $\gamma$-ray instruments such as AGILE, INTEGRAL/SPI-ACS, Insight/HXMT,
and Konus-Wind et al. (2019). MAGIC detected GRB 190114C starting $\sim 60$ s
after $T_{0}$ with a significance above $50\sigma$.

####  4.3.1 General properties and multi-wavelength observations

The duration of the prompt emission is $T_{90}\approx 116$ s as measured by
Fermi-GBM and $T_{90}\approx 362$ s by Swift-BAT. However, the prompt light
curve showed a multi-peak structure only for about 25 s, suggesting that the
remaining activity, which is characterised by a smooth power-law decay,
recorded by these instruments may be already the afterglow emission. Support
to such interpretation is also obtained from a joint spectral and temporal
analysis of the Fermi-GBM and Fermi-LAT data Ravasio et al. (2019). The total
radiated prompt energy is $E_{\gamma,iso}=(2.5\pm 0.1)\times 10^{53}$ erg in
the energy range 1–104 keV Ajello (2020).

Figure 17: GRB 190114C: light-curves at different frequencies. From et al.
(2019)

Extensive follow-up observations from several different instruments from GeV
to radio are available. Light-curves are shown in Figure 17. Fermi-LAT
observations started since the beginning of the prompt phase. A GeV
counterpart was detected from $T_{0}$ to $150$ s, when the burst left the LAT
field of view and remained outside it until $8600$ s. When LAT resumed
observations significant signal was still detected at a flux level $\sim
2\times 10^{-10}$ erg cm-2 s-1 (0.1-1 GeV). After $\sim 60$ s from the burst
trigger, Swift-XRT started follow-up observations, which covered in total
$\sim 10^{6}$ s. The light-curve in the 1-10 keV energy band is consistent
with a power-law decay $F\propto t^{\alpha}$ with $\alpha=-1.36\pm 0.02$ et
al. (2019). NuSTAR and XMM-Newton observations are also available around 1-2
days. The NIR, optical and UV data were taken from around $\sim 100$ s. The
early emission is particularly bright and is interpreted as dominated by the
reverse shock component Laskar et al. (2019). Afterwards, the decay rate
flattens and then steepens again after $\sim 3\times 10^{4}$ s (see Figure
17). The Nordic Optical Telescope measured a redshift of $z=0.4245\pm 0.0005$
Selsing et al. (2019) which was then confirmed by Gran Telescopio Canarias
Castro-Tirado et al. (2019). Radio and sub-mm data were taken from $\sim
10^{4}$ s and exhibit an achromatic behavior, possibly dominated by the
reverse shock in the sub-mm range, followed by emission at late times with
nearly constant flux.

####  4.3.2 VHE observations and results

Figure 18: GRB 190114C. Upper panel: MAGIC light-curve (red circles), compared
with the XRT (green band) and LAT (red band) emission. Bottom panel: temporal
evolution of the intrinsic spectral photon index in the MAGIC data analysis
time bins. From MAGIC Collaboration et al. (2019).

After receiving (at 22 s after the BAT trigger time) and validating (at 50 s)
the GRB alert, the MAGIC telescopes started observing GRB 190114C at 57 s and
operated stably from 62 s, starting from a zenith angle of 55.8∘. Observations
lasted until 15912 s when a zenith angle of 81.14∘ was reached. The
observation was performed in good weather conditions but in presence of the
Moon, resulting in a night sky background approximately 6 times higher than
the standard dark night conditions. The results of the offline analysis show a
clear detection above the $50\sigma$ level in the first 20 minutes of
observation MAGIC Collaboration et al. (2019).

The light-curve (see Figure 18, upper panel) for the intrinsic flux (i.e.,
corrected for the EBL absorption) in the 0.3–1 TeV range was derived starting
from 62 s and up to 2454 s. The TeV light curve is well described by a power-
law with temporal decay index $\beta_{T}=-1.60\pm 0.07$, steeper than the one
exhibited by the X-ray flux. The temporal evolution of the intrinsic spectral
photon index $\alpha_{int}$ of the TeV differential photon spectrum is shown
in the bottom panel. A constant value of $\alpha_{int}$ $\approx-2$ is
consistent with the data, considering the statistical and systematic errors,
but there is evidence for a softening of the spectrum with time. The spectral
fit in the 0.2-1 TeV energy range for the time-integrated emission ($62-2454$
s) returns $\alpha_{obs}=-5.34\pm 0.22$ and
$\alpha_{\textup{int}}=-2.22^{+0.23}_{-0.25}$ for the observed and EBL-
corrected spectrum, respectively.

####  4.3.3 Interpretation

Figure 19: GRB 190114C: comparison between the distribution of the observed
$\gamma$-ray events binned in time and energy (blue shaded areas) and the
limiting curves for synchrotron maximum energy. Two different density radial
profiles are considered for the derivation of the theoretical curves: constant
(red dotted curve) and wind-like (red dashed curve). From MAGIC Collaboration
et al. (2019).

The properties of the VHE light curve and spectrum of GRB 190114C were studied
by the MAGIC Collaboration in MAGIC Collaboration et al. (2019). The PL
behaviour, the absence of variability, and the relatively long timescale of
the emission support the evidence that the VHE component belongs to the
afterglow phase. An estimate of the amount of radiated power in the TeV range
can be derived, assuming that the afterglow onset is at $\sim 6$ s Ravasio et
al. (2019). In this case the energy radiated in the TeV band is $\sim 10\%$ of
the isotropic-equivalent energy of the prompt emission $E_{\gamma,iso}$
considering the temporal evolution estimated from the MAGIC light-curve.

The energy of the photons observed by the MAGIC telescopes was compared with
the maximum energy of synchrotron photons assuming two possible scenarios for
the radial profile of the external density, namely constant and wind-like
(Figure 19). These estimates of the maximum energy are based on the widely
adopted limit on the maximum electron Lorentz factor set by equating the
acceleration at Bhom rate with the synchrotron cooling rate (see Eq. 57).
Adopting this limit, synchrotron emission can not account for the TeV photons
detected by MAGIC, and a different radiation mechanism must be invoked.

An additional, model independent indication for the presence of a spectral
component other than synchrotron is evident after multi-wavelength
simultaneous SEDs are built. In et al. (2019) the VHE data were rebinned into
five time intervals and XRT, BAT, GBM and LAT data were added when available,
i.e., in the first two time intervals (Figure 20). The spectrum shows a
double-peaked behaviour with a first peak in the X-ray band and the second one
in the VHE band. The Fermi-LAT data play a particularly important role in
revealing the shape of the SED, as they show a dip in the flux, strongly
supporting an interpretation of the whole SED as superposition of two distinct
components.

Figure 20: GRB 190114C SEDs from soft X-rays to 1 TeV in five different time
intervals. The contour regions for flux uncertainties are also shown. For
MAGIC and LAT the 1$\sigma$ error of their best-fit power-law functions are
displayed. For Swift data, the 90% confidence contours for the XRT-BAT joint
fit is shown. MAGIC spectra are EBL-corrected assuming the model by Domínguez
et al. (2011). From et al. (2019).

Following these considerations, in et al. (2019) the SSC mechanism is
explored. The broad-band emission is modeled with a numerical code reproducing
the synchrotron and SSC radiation in the external forward shock scenario,
including the proper KN cross section and the effects of $\gamma-\gamma$
annihilation.

The predicted spectra and lightcurves are compared with the data in Figure 21
and Figure 22. Acceptable modeling of the multi-wavelength afterglow spectra
have been found for a constant medium with $E_{k}\gtrsim 3\times 10^{53}$ erg,
$\varepsilon_{e}\approx 0.05-0.15$, $\varepsilon_{B}\approx(0.05-1)\times
10^{-3}$, $n\approx 0.5-5$ cm-3 and $p\approx 2.4-2.6$. It is found that,
being the peak of the SSC component below 200 GeV, the KN suppression and the
$\gamma-\gamma$ internal absorption have a non-negligible role in shaping the
peak of the VHE spectrum. The modeling reproduces very well the XRT, LAT and
TeV emission (solid blue curve in Figure 21 and solid blue, green and red
curves in Figure 22), while it overproduces both the optical and radio flux at
late times (solid violet, yellow and cyan curves in Figure 22). According to
et al. (2019), a similar fit is found also assuming a wind-like profile for
the external density. In this case the parameters are $E_{k}=4\times 10^{53}$
erg, $\varepsilon_{e}=0.6$, $\varepsilon_{B}=1\times 10^{-4}$, $A_{\ast}=0.1$
and $p=2.4$. Very interestingly, the modeling shows that the late LAT
observation (around $10^{4}$ s) is completely dominated by SSC emission (red
dashed curve in Figure 22). A different type of modelization is also
investigated by et al. (2019), under the requirement to model optical data. In
this case (dotted curves in Figure 22), the fit is very good for optical,
X-ray and LAT observations, but fails in reproducing the MAGIC light-curve.

Figure 21: GRB 190114C: modeling of two SEDs in consecutive time intervals.
The different curves refer to: the observed spectrum (thin solid line), the
EBL deabsorbed spectrum (thick blue line) and the SSC component neglecting the
effects of internal $\gamma$-$\gamma$ opacity (dashed line). From et al.
(2019).

The values inferred for the GRB afterglow parameters are similar to those used
for past GRB afterglow studies at lower frequencies. This is an indication
that the SSC component can be a relatively common process for GRB afterglows,
since it does not require peculiar values of the parameters to be explained.

Figure 22: Modeling of broad band GRB 190114C light curves. Description of the
different modelings used is given in Section 19. From et al. (2019).

Several other successful modelings of GRB 190114C data within the synchrotron
and SSC external forward shock scenario have been published in literature Wang
et al. (2019); Derishev and Piran (2021); Asano et al. (2020); Joshi and
Razzaque (2021). A summary of the parameters inferred by different works can
be found in Table 2.

In Wang et al. (2019) the X-ray, optical and LAT data before 100 s are
attributed to reverse shock emission or prompt contribution. A constant-
density environment for the circumburst material is assumed. A time-averaged
SED ($50-150$ s) is estimated showing that at GeV energies a transition
between the synchrotron and SSC component can be identified. From re-analysis
of LAT data, a hard photon index ($1.76\pm 0.21$) is derived, in agreement
with the hardening of the spectrum caused by the rising of the SSC component.
Differently from what seen by et al. (2019); Derishev and Piran (2021)
$\gamma-\gamma$ absorption does not contribute significantly in shaping the
VHE spectrum.

A similar interpretation is given in Asano et al. (2020), although the
inferred value of $\epsilon_{B}$ is larger ($\epsilon_{B}\sim 10^{-3}$). A
consistent modeling of the multi-wavelength observations as synchrotron and
SSC radiation is found, both the ISM and wind-like scenarios. The SED at 80 s
(see figure 2 and figure 5 in Asano et al. (2020)) and the broad-band light
curves (figure 3 and figure 6) are reproduced, despite at 103 s, the model
predictions in the $0.3-1$ TeV band and X-rays are slightly brighter than the
observed data.

In Joshi and Razzaque (2021), analytical approximations are adopted for the
description of the synchrotron and SSC components. In addition, the KN cut-off
energy and the $\gamma-\gamma$ absorption contribution are calculated and
compared with the data. A wind-like environment was used for the circumburst
medium. The SEDs in the time intervals $68-110$ s and $110-180$ s are
modelled, and the two values of the KN cut-off energies calculated at these
times are $\sim 3.7$ TeV and $2.1$ TeV. This implies that the KN effect is
relevant only at TeV levels and the VHE data can be modelled assuming that the
SSC scattering is in Thompson regime. The $\gamma-\gamma$ absorption is also
considered negligible since the estimated attenuation factor is way lower than
the one due to EBL attenuation and it reaches values around unity only for
energies $\gtrsim 1$ TeV.

In Derishev and Piran (2021) the multi-wavelength data were fitted with a
single-zone numerical code with an exact calculation of KN cross-sections as
well as the attenuation due to the pair production mechanism. A smoothed
analytical approximations for the electron injection function was used. A
systematic scan over a 4-dimensional parameter space was performed to search
for the best-fit solution at early and later times. The SED calculated at 90 s
and 150 s (figure 3 and figure 9, respectively) are found to be well described
by a fast cooling regime. The KN effect and the pair production mechanism
shape the VHE spectrum significantly. It is estimated that $\simeq 10\%$ of
the total emitted power, i.e., $\simeq 25\%$ of initially produced IC power,
is absorbed.

$E_{k}$ $\epsilon_{e}$ $\epsilon_{B}$ n $p$ $\xi_{e}$ erg cm-3 MAGIC Coll.
$\gtrsim 3\times 10^{53}$ 0.05-0.15 0.05-1 $\times 10^{-3}$ 0.5-5 2.4-2.6 1
Wang + 2019 $6\times 10^{53}$ 0.07 $4\times 10^{-5}$ 0.3 2.5 1 Asano + 2020
$10^{54}$ 0.06 $9\times 10^{-4}$ 1 2.3 0.3 Asano + 2020 $10^{54}$ 0.08
$1.2\times 10^{-3}$ 0.1 (wind) 2.35 0.3 Joshi + 2021 $4\times 10^{54}$ 0.03
0.012 $2\times 10^{-2}$ (wind) 2.2 1 Derishev + 2021 $3\times 10^{53}$ 0.1
$2-6\times 10^{-3}$ 2 2.5 1

Table 2: GRB 190114C: parameters inferred by different authors from the
modeling of observations with a synchrotron-SSC scenario.

###  4.4 GRB 190829A

GRB 190829A is a nearby ($z=0.078$) long GRB triggered by Swift-BAT Dichiara
et al. (2019) and Fermi-GBM Fermi GBM Team (2019). The Fermi-GBM trigger time
is $T_{0}=19:55:53.13$ UTC. H.E.S.S. detected this GRB over three consecutive
nights, with significance of $21.7\sigma$ during the first night ($\sim 4$ h
after the GRB trigger).

####  4.4.1 General properties and multi-wavelength observations

The prompt emission detected by the two instruments consists of two episodes
with the first one seen in the time interval from the trigger time to 4 s and
the second brighter episode from 47 s to 61 s. The two episodes have very
different spectral properties: the first one is described by a power-law with
index $-1.41\pm 0.08$ and an exponential high-energy cutoff function with
$E_{p}=130\pm 20$ keV, the second one can be described with a Band function
with $E_{p}=11\pm 1$ keV , $\alpha=-0.92\pm 0.62$ and $\beta_{T}=-2.51\pm
0.01$ Chand et al. (2020). The (isotropic equivalent) prompt emission energy
inferred from spectral analysis of Fermi-GBM data is $E_{\gamma,iso}\sim
2\times 10^{50}$ erg.

A multi-wavelength observational campaign of the event was performed covering
the entire electromagnetic spectrum (see Figure 23 and 25). The event was not
detected in the HE range by Fermi-LAT. Nevertheless, ULs have been reported in
the MeV-GeV band up to $3\times 10^{4}$ s Piron et al. (2019). The Swift-XRT
started observations at 97.3 s and detected a bright X-ray afterglow, which
was monitored until $\sim 7.8\times 10^{6}$ s Osborne et al. (2019). The X-ray
light curve in the 0.3-10 keV energy range (observer frame) shows a peculiar
behaviour with an initial steep decay phase followed by a plateau and a strong
flare episode (Figure 25, upper panel, blue points). After the flare the
standard afterglow phase starts with a decay following a power-law with a
possible steepening around 10 days. In the UV/optical/NIR band the event was
followed by several instruments. The redshift was estimated to be $z=0.0785\pm
0.005$ Valeev et al. (2019), which makes this event one of the closest GRBs
ever detected. Starting from 4.5 - 5.5 days after the GRB trigger an
associated supernova has been reported Lipunov et al. (2019). Also in the
optical data a flare is seen simultaneously with the one in X-rays. In the
radio band the detection was reported by several instruments starting from
$\sim 1$ day after the trigger Laskar et al. (2019); Monageng et al. (2019);
de Ugarte Postigo et al. (2019); Chandra and et al. (2019). The radio flux
initially slowly increases and then starts to decay after 20-30 days.

####  4.4.2 VHE observations and results

Figure 23: GRB 190829A: multi-wavelength light curves (A) (upper panel) and
photon index evolution (bottom panel) in the X-ray, HE and VHE band (B). The
BAT prompt light curve is shown in the inset (C). From H. E. S. S.
Collaboration et al. (2021).

The afterglow emission of GRB 190829A was followed-up by the H.E.S.S.
telescopes starting at 4.3 h and continued for three consecutive nights (up to
55.9 h). Observations were performed using the four medium-size telescopes of
H.E.S.S. for a total amount of 13 h divided respectively in 3.6 h (starting at
4.3 h), 4.7 h (starting at 27.2 h) and 4.7 h (starting at 51.2 h). The
statistical significance at the GRB position found in the three nights are
respectively $21.7\sigma$, $5.5\sigma$ and $2.4\sigma$.

Spectral analysis was performed for the first two nights fitting the observed
photon spectrum with a power-law model. The following values are found:
$\alpha_{obs}=-2.59\pm 0.09$ (stat.) $\pm 0.23$ (syst.) in the $0.18-3.3$ TeV
(first night) and $\alpha_{obs}=-2.46\pm 0.22$ (stat.) $\pm 0.14$ (syst.) in
the 0.18 - 1.4 TeV energy range (second night). Fitting a power-law attenuated
by EBL, the photon indices inferred for the intrinsic spectrum are:
$\alpha_{int}=-2.06\pm 0.10$ (stat.) $\pm 0.26$ (syst.) in the $0.18-3.3$ TeV
energy range (first night) and $\alpha_{int}=-1.86\pm 0.26$ (stat.) $\pm 0.17$
(syst.) in the 0.18 - 1.4 TeV energy range (second night). The photon indices
in each night are consistent, within systematical uncertainties, with those of
the simultaneous X-ray emission. Combining all three nights, the photon index
is $\alpha_{int}=-2.07\pm 0.09$ (stat.) $\pm 0.23$ (syst.) in the 0.18 - 3.3
TeV energy range.

The light-curve in the 0.2 - 4.0 TeV energy range derived up to 56 h is
compared in Figure 23 with the XRT light-curve and the LAT upper limits. The
time-evolving flux was satisfactorily modeled with a power-law decay
$F(t)\propto t^{\alpha}$ with $\alpha=-1.09\pm 0.05$. Such decay index is
similar to the X-ray one derived in the same time interval
($\alpha_{X}=-1.07\pm 0.09$).

####  4.4.3 Interpretation

The interpretation of the VHE emission from GRB 190829A is debated and
different radiation mechanisms including synchrotron, SSC or EIC emission have
been proposed so far to explain the origin of the TeV emission.

Figure 24: GRB 190829A: modeling of X-ray, LAT and H.E.S.S. data proposed by
the H.E.S.S. collaboration for the two time intervals with VHE and X-ray
detections. Two scenarios are investigated for the TeV emission: synchrotron
and SSC. H.E.S.S. flux contours are displayed considering the statistical
uncertainty. The synchrotron and SSC component are shown in dashed and dash-
dotted lines respectively. The shaded areas represent the 68$\%$ confidence
intervals determined from the posterior probability distribution of the MCMC
parameter fitting for the standard SSC model (light blue) and for the model
without maximum energy for synchrotron emission (orange). From H. E. S. S.
Collaboration et al. (2021).

The H.E.S.S. Collaboration H. E. S. S. Collaboration et al. (2021)
investigated both the synchrotron and the SSC emission in the external forward
shock as responsible radiation mechanism of the observed TeV component. Multi-
wavelength data collected simultaneously with H.E.S.S. observations in the
first two nights were modeled separately with a time-independent numerical
code using the Markov-chain Monte Carlo (MCMC) approach to explore the
parameter space. The results of the fitting show that the SSC mechanism fails
to explain the VHE emission. The low Lorentz bulk factor predicted by the
observations ($\Gamma\lesssim 10$) implies that the SSC emission occurs in KN
cross-scattering regime. As a result, a steep spectrum, inconsistent with the
observational VHE data, is obtained (see Figure 24, light blue shaded area).
Possible improvements between the data and the model foresee a higher
$\Gamma$, which is in contrast with the observations, or the presence of an
additional hard component in the distribution of the accelerated electrons.
However, this latter solution implies extreme assumptions on the density of
the circumburst medium ($n_{0}=10^{-5}$ cm-3 in case of strong magnetic field
or $n_{0}=10^{5}$ cm-3 for weak magnetic field) and a SED strongly dominated
by the SSC component which is inconsistent with the data. A better fitting of
the observational data can be obtained when considering an alternative model
where the maximum electron energy set by the radiative losses is ignored. In
such scenario, the synchrotron emission is able to extend up to TeV energies
and the observational broad-band data are described by a single synchrotron
component (see Figure 24, orange shaded area). The SSC contribution is
negligible while the $\gamma-\gamma$ absorption shapes the VHE spectrum. The
single synchrotron component scenario provides better fit ($>5\sigma$) to the
multi-wavelength data. On the other hand, this interpretation requires unknown
acceleration processes or non-uniform magnetic field strength in the emission
region as described for GRB 180720B (see Section 4.2).

Figure 25: GRB 190829A: modeling of multi-wavelength light curves (upper
panel) and SED (bottom panel) proposed by Salafia et al. (2021). The 90$\%$
and 50$\%$ credible intervals from the fit are shown in lighter shades.

A complete multi-wavelength modeling of the GRB 190829A data including
contribution of the synchrotron and SSC emission for both the forward and
reverse shocks considering a constant-density environment is presented in
Salafia et al. (2021). The predicted broad-band light curves and the SED at
the time of the H.E.S.S. detection are shown in Figure 25. A MCMC approach was
adopted in order to estimate the best-fit parameters for the multi-wavelength
modeling. The resulting values of the parameters related with the forward
shock scenario are shown in Table 3. In contrast with the H.E.S.S.
Collaboration results, the VHE emission is well reproduced with the SSC
external forward shock scenario. The usual simplified assumption that
$\xi_{e}=1$ is excluded by the fit which provides acceptable solutions only
for $\xi_{e}\lesssim 6.5\times 10^{-2}$. Moreover, an isotropic-equivalent
kinetic energy at the afterglow onset $E_{k}=2.5^{+1.9}_{-1.3}\times 10^{53}$
erg is estimated. Considering the observed GBM prompt energy, such value
implies that the prompt efficiency is $\eta=1.2^{+1.0}_{-0.5}\times 10^{-3}$
which is much lower than the typical values derived from previous GRB studies.
The other parameters ($n_{0}$, $\epsilon_{e}$ and $\epsilon_{B}$) are found to
be similar to the ones estimated for GRB 190114C.

A two-component off-axis jet model has also been investigated Sato et al.
(2021). Such model proposes that the GRB jet is seen off-axis
($\theta_{view}=1.78^{\circ}$) and it consists of a narrow
($\theta_{jet}=0.86^{\circ}$) fast ($\Gamma=350$) jet and a slow ($\Gamma=20$)
co-axial jet. The former jet component is responsible for the emission of SSC
photons in the VHE band. The calculation of the SSC flux at the time of the
H.E.S.S. detection is done following the prescriptions of Sari, Re’em and Esin
(2001) considering only the Thompson scattering regime.

An EIC plus SSC scenario has also been proposed for the production of the VHE
component Zhang et al. (2021). The seed photons belong to the long-lasting
X-ray flare seen for GRB 190829A which can be up-scattered to TeV energies. A
numerical calculation of the afterglow dynamics and radiative processes have
been used to model the observational data. For $t\sim 10^{3}-10^{4}$ s the EIC
component dominates the VHE emission, while for later times ($t\gtrsim 3\times
10^{4}$ s) the EIC gradually decays and the SSC component becomes relevant.
The initial afterglow kinetic energy used for the modeling ($E_{k}=10^{52}$
erg) suggests that GRB 190829A is not a typical low-luminosity GRB but it may
have much higher kinetic energy.

$E_{k}$ $\epsilon_{e}$ $\epsilon_{B}$ n $p$ $\xi_{e}$ $\theta_{j}$ erg cm-3
rad Hess Coll. (SSC) $2.0\times 10^{50}$ 0.91 $5.9-7.7\times 10^{-2}$ 1.
2.06-2.15 1. / Hess Coll. (Sync) $2.0\times 10^{50}$ 0.03-0.08 $\approx 1$ 1.
2.1 1. / Salafia + 2021 $1.2-4.4\times 10^{53}$ 0.01-0.06 $1.2-6.0\times
10^{-5}$ 0.12-0.58 2.01 $<6.5\times 10^{-2}$ 0.25-0.29 Zhang + 2021 $9.8\times
10^{51}$ 0.39 $8.7\times 10^{-5}$ 0.09 2.1 0.34 0.1

Table 3: Parameters for modeling of GRB 190829A

###  4.5 GRB 201015A

GRB 201015A is a long GRB at $z=0.426$ detected by the Swift-BAT D’Elia et al.
(2020) on 15 October 2020, at $T_{0}=22:50:13$ UT. The Fermi-GBM instrument
did not trigger the event but the targeted search revealed a transient source
consistent with the Swift-BAT location Fletcher et al. (2020). MAGIC
observations show a possible detection with a significance of $\sim
3.5\sigma$.

####  4.5.1 General properties and multi-wavelength observations

The (isotropic equivalent) prompt emission energy inferred from spectral
analysis of Fermi-GBM data is $E_{\gamma,iso}=(1.1\pm 0.2)\times 10^{50}$ erg
Minaev and Pozanenko (2020). The prompt duration is $T_{90}=9.78\pm 3.47$ s
($15-350$ keV band). The BAT time-average spectrum in the time interval $0-10$
s is well fitted by a power-law model with photon index $\beta_{T}=-3.03\pm
0.68$ suggesting a low peak energy $E_{p}<10$ keV Markwardt et al. (2020).

Swift-XRT D’Ai et al. (2020) follow-up the event starting only 3214 s after
$T_{0}$ due to observational constraints. The light curve up to almost 1 day
is well described by a power-law with decay index
$\alpha=-1.49^{+0.24}_{-0.21}$. Late-time observations performed by the
Chandra X-ray Observatory Gompertz et al. (2020) and Swift-XRT D’Elia and
Swift Team (2020) from $\sim 8$ days up to $\sim 21$ days showed a flattening
of the X-ray light curve, i.e. a flux level inconsistent (higher) with the
extrapolation of the power-law decay rate at early times. Optical observations
confirmed the presence of an afterglow counterpart from around 168 s Lipunov
et al. (2020). The optical light-curves showed a clear initial rise with a
peak around $200$ s followed by a decay Grossan et al. (2020). A bright radio
counterpart (flux density $\sim 1.3\times 10^{-4}$ Jy at 6 GHz 1.4 days after
the burst) was also detected by several instruments Fong et al. (2020);
Giarratana et al. (2020); Marcote et al. (2020). Late-time optical
observations identified an associated supernova rising from 5 days after the
burst reaching its maximum flux around 12-20 days after $T_{0}$ Pozanenko et
al. (2020); Rossi et al. (2021). The measurement of the redshift was reported
by the GTC ($z=0.426$) de Ugarte Postigo et al. (2020) and then confirmed by
the NOT ($z=0.423$) Izzo et al. (2020) instrument.

####  4.5.2 VHE observations and results

Figure 26: GRB 201015A: distribution of the angular distance $\theta^{2}$
between the reconstructed event arrival directions and the nominal source
position. The gray histogram represents background events while the black
point with blue crosses are the $\gamma$-like events. The vertical dashed line
describes the $\theta^{2}$ cut value and defines the region in which excess
events and signal significance are calculated. From Suda et al. (2021).

Final results from VHE data analysis of GRB 201015A have not been published
yet. Preliminary information reported here have been released in Blanch et al.
(2020) and Suda et al. (2021). Observations of GRB 201015A were performed by
the MAGIC telescopes starting 33 s after the trigger time, under dark
conditions, with a zenith angle ranging from 24∘ up to 48∘, and lasted for
about 4 h. In the second half of the data taking the presence of passing
clouds affected the observation for $\sim$0.45 h. These data were removed and
the remaining ones were analyzed with the standard MAGIC analysis software.
Offline analysis showed a possible excess with a $3.5\sigma$ significance at
the GRB position (see Figure 26) and a significant spot in the sky map. The
energy threshold of the analysis is calculated to be 140 GeV from Monte Carlo
simulated $\gamma$-ray data.

###  4.6 GRB 201216C

GRB 201216C is a long GRB at $z=1.1$ triggered by the Swift-BAT at
$T_{0}=23:07:31$ UT on 16 December 2020 Beardmore et al. (2020). Fermi-GBM
also detected the event with a slightly different trigger time (6 second
before the Swift-BAT) Malacaria et al. (2020). MAGIC detected GRB 201216C with
a significance of $\sim 6\sigma$.

####  4.6.1 General properties and milti-wavelength observations

The duration is estimated as $T_{90}=48\pm 16$ s in the 15-350 keV band by
Swift-BAT Ukwatta et al. (2020) and around $29.9$ s in the 50-300 keV band by
Fermi-GBM. The light curve shows a multiple peak structure with a main peak
around 20 s after the trigger time. The time-averaged GBM spectrum in the
first 50 s is best fit by a Band function with $E_{p}=326\pm 7$ keV,
$\alpha=-1.06\pm 0.01$ and $\beta_{T}=-2.25\pm 0.03$. The isotropic equivalent
energy $E_{\gamma,iso}$ in the 10-1000 keV band is $(4.71\pm 0.16)\times
10^{53}$ erg, as calculated from the fluence measured by Fermi-GBM.

Fermi-LAT observed the GRB starting from around 3500 s and up to 5500 s. No
significant emission was reported Bissaldi et al. (2020). Swift-XRT began the
observation at $t=2966.8$ s due to an observational constrain. A fading source
was detected following a broken power-law behaviour with decay indices of
$1.97^{+0.10}_{-0.09}$ and $1.07^{+0.15}_{-0.10}$ and a break at 9078 s Evans
et al. (2021). Optical observations were also performed by several
instruments. The $r$-band light curve made along with VLT data point Izzo et
al. (2020) and inferred data from FRAM-ORM Jelinek et al. (2020) show a power-
law decay in flux with index equal to 1. The Liverpool Telescope observations,
performed around 177 s after the trigger time, seems to be around the peak of
the optical afterglow Shrestha et al. (2020). The HAWC observatory followed-up
the event but no significant detection was identified in the TeV band Ayala
(2020). Redshift estimation of $z=1.1$ was performed by the ESO VLT Vielfaure
et al. (2020).

####  4.6.2 VHE observations and results

Figure 27: $\theta^{2}$ angular distance distribution between reconstructed
event arrival directions and nominal source position for GRB 201216C. The gray
histogram represents background events while the black point with blue crosses
are the $\gamma$-like events. The vertical dashed line describes the
$\theta^{2}$ cut value and defines the region in which excess events and
signal significance are calculated. From Fukami et al. (2021).

Final results from VHE data analysis of GRB 201216C have not been published
yet. Preliminary information reported here have been released in Blanch et al.
(2020) and Fukami et al. (2021). MAGIC observations and data taking of GRB
201216C started with a delay of 56 s after the Swift-BAT trigger time. The
observation lasted for 2.2 h and was performed in optimal atmospheric
condition and in absence of Moon. The zenith angle ranged from 37∘ to 68∘. The
low level of night sky background allowed to retain also the low energy events
and therefore obtain a low energy threshold with compared to the other GRBs
observed. To keep as many low-energy events as possible, an image cleaning
method to extract dimmer Cherenkov showers initiated by gamma rays than the
standard method was adopted.

The signal significance was calculated to be $6.0\sigma$ pre-trial
($5.9\sigma$ post-trialA trial factor of 2 is considered due to the two sets
of analysis cuts used for MAGIC data analysis.) for the first 20 minutes of
observation (see Figure 27). A preliminary time-integrated spectrum for the
first 20 minutes of observation was produced. Due to the strong absorption
effect by EBL a very steep power-law decay was found for the observed
spectrum, especially for the events with energies higher than a few hundreds
of GeV. The intrinsic spectrum, corrected for the EBL absorption was found to
be consistent with a flat single power-law until 200 GeV above which no
significant spectral points have been derived. A preliminary light curve in
the time interval 56 s - 2.2 h was also calculated. After 50 min only upper
limits on the emitted flux have been derived since no significant emission was
found after this time. The preliminary results are consistent with a
monotonically decaying light curve fitted with a power-law.

##  5 The new TeV spectral window: discussion

After decades of searches, MAGIC and H.E.S.S. observations have unequivocally
proved that (long) GRBs can be accompanied by a significant amount of TeV
emission during the afterglow phase. Table 4 summarizes the main properties of
the GRBs detected by IACTs, and presented in detail in the previous section.
The list includes also two events (namely GRB 160821B and GRB 201015A) where
only a hint of excess (i.e., with a significance at $\sim 3-4\sigma$) was
found. For the other four events, namely GRB 180720B, GRB 190114C, GRB 190829A
and GRB 201216C the detections are robust ($>5\sigma$). The table lists
several properties, such as duration $T_{90}$ and total emitted energy
$E_{{\gamma,iso}}$ of the prompt emission, redshift, and information on the
IACT detection (the starting time $T_{delay}$ of observations elapsed since
the trigger time $T_{0}$, the energy range where photons have been detected,
the name of the telescope and the significance of the excess). GRB 160821B is
the only one belonging to the short class, the other five being long GRBs.

In this section we address the question why these GRBs have been detected,
whether they have peculiar properties and whether they show some common
behaviour which may be at the basis of the production of TeV radiation. To do
that, one should be careful, since these GRBs have been followed-up under very
different observational conditions and with very different time delays after
the trigger time, and they span a quite large range of redshifts (from 0.078
to 1.1). Keeping in mind these differences, which have a strong impact on the
detection capabilities of IACTs, we compare the observed and intrinsic
properties of the population of GRBs at VHE, highlighting their similarities
and differences, and discuss how they compare to the whole population.

$T_{90}$ $E_{\gamma,iso}$ z $T_{delay}$ $E_{range}$ IACT (sign.) s erg s TeV
160821B 0.48 $1.2\times 10^{49}$ 0.162 24 0.5-5 MAGIC ($3.1\sigma$) 180720B
48.9 $6.0\times 10^{53}$ 0.654 3.64$\times 10^{4}$ 0.1-0.44 H.E.S.S.
($5.3\sigma$) 190114C 362 $2.5\times 10^{53}$ 0.424 57 0.3-1 MAGIC
($>50\sigma$) 190829A 58.2 $2.0\times 10^{50}$ 0.079 1.55 $\times 10^{4}$
0.18-3.3 H.E.S.S. ($21.7\sigma$) 201015A 9.78 $1.1\times 10^{50}$ 0.42 33 0.14
MAGIC ($3.5\sigma$) 201216C 48 $4.7\times 10^{53}$ 1.1 56 0.1 MAGIC
($6.0\sigma$)

Table 4: List of the GRBs observed by IACTs with a firm detection
(significance $>5\sigma$) or a hint of detection ($3-4\sigma$) above 100 GeV.
The $T_{90}$ and $E_{\gamma,iso}$ refer to the duration and total emitted
energy of the prompt emission; the redshift is listed in column 3; $T_{delay}$
is the time delay between the trigger time $T_{0}$ and the time when IACT
observations started; $E_{range}$ defines the energy range of the detected
photons. The name of the telescope which made the observation and the
significance of the detection are listed in the last column.

###  5.1 Observing conditions

Low zenith angles, fast repointing, dark nights, low redshift, and highly
energetic events have always been considered as optimal, if not necessary,
conditions to have some chances for GRB detections with IACTs. On the other
hand, these first VHE GRBs have demonstrated that GRBs can have a level of TeV
emission large enough to be detected by the current generation of IACTs, even
under non-optimal conditions. GRB 190114C was observed with a zenith angle
$>55^{\circ}$ and in presence of the Moon. Both conditions imply a higher
energy threshold (typically $\gtrsim$ 0.2 TeV) and require a dedicated data
analysis. Another example is GRB 160821B, that was observed with a NSB 2-8
times higher than the standard dark night conditions. Moreover, significant
VHE excess was found not only in case of short delays (less than hundreds of
seconds) from the burst trigger but, somewhat surprisingly, also at quite late
times, i.e. with delays of several hours or even days, as in the case of GRB
180720B and GRB 190829A, respectively. This showed the importance of pointing
a GRB also at relatively late times, in cases fast follow-up observations are
not feasible.

Optimal observing conditions and short delays remain however crucial to detect
GRBs at higher redshift, for which the impact of EBL is large already at a few
hundreds GeV. This explains how it has been possible the detection of a GRB at
$z=1.1$ (GRB 201216C): in this case, optimal observing conditions allowed to
reach a low energy threshold of the sensitivity window ($\sim 70$ GeV). The
excess of signal was indeed found only below 200 GeV (more precisely, between
70 and 200 GeV) where the attenuation by the EBL is still limited.

###  5.2 Redshift and the impact of EBL

The redshift of the detected GRBs covers a broad range, from $z=0.079$ (GRB
190829A) to $z=1.1$ (GRB 201216C). The impact of the EBL attenuation on the
spectrum is severely changing, depending on the redshift value and on the
photon energy. For redshift $z\sim 0.4$ the impact becomes relevant for
energies $\gtrsim 0.2$ TeV with a flux attenuation of $\sim 50\%$ for $0.2$
TeV and almost $\sim 99.5\%$ for $1$ TeV Domínguez et al. (2011). For nearby
events ($z\lesssim 0.1-0.2$) the effect of EBL is less severe and becomes
relevant only for energies $\gtrsim 0.3$ TeV, reaching an attenuation factor
of an order of magnitude only for energies $\gtrsim 2$ TeV. As a result, the
GRB observed photon indices and the energy range of detected TeV photons
differs significantly between the events. GRBs with redshift $z>0.4$ such as
GRB 190114C or GRB 180720B have very steep photon indices and they are
detected in the lower energy range up to 0.44 TeV for GRB 180720B and 1.0 TeV
for GRB 190114C. Spectral analysis from GRB 201216C are not yet public but
preliminary results indicate that the emission is concentrated in the lower
energy band between 0.1-0.2 TeV. Nearby GRBs with redshift $z\lesssim 0.1-0.2$
such as GRB 160821B or GRB 190829A show a less steep photon spectrum (around
$-2.5$) and the TeV detection range extends above 1 TeV. The detection of
several GRBs with significant value of redshift ($z>0.4$) is a robust proof
that IACTs can overcome the limitations due to the EBL absorption and can
expand the VHE detection horizon at the current stage up to $z=1.1$. On the
other hand, it is evident that detection of nearby GRBs is fundamental in
order to explore more robustly the spectral shape, unbiased by the EBL effect
which is a non-negligible source of uncertainty for higher redshifts.

Figure 28: The Amati ($E_{peak}$ \- $E_{iso}$) correlation for a sample of 136
long GRBs (grey dots, from Nava et al. (2012)) and a sample of 11 short GRBs
(empty blue squares) detected by Swift. The corresponding power-law fit for
the sample of long GRBs and the $3\sigma$ scatter of the distribution of
points around the best fits are shown. The six GRBs detected in VHE are also
added in the plot. Adapted from D’Avanzo et al. (2014).

###  5.3 Energetics

In terms of $E_{\gamma,iso}$ the VHE GRB sample spans more than three orders
of magnitude from $\sim 10^{49}$ erg up to $\sim 6\times 10^{53}$ erg. The
five long GRBs detected follows the Amati correlation as shown in Figure 28.
GRB 160821B, the only short GRB of the sample, is consistent with the
existence of a possible Amati-like correlation for short GRBs, with this event
falling in the weak-soft part of the correlation. The detections of GRB
190829A and GRB 201015A show that an event does not need to be extremely
energetic in terms of isotropic-equivalent prompt energy in order to produce a
TeV emission with (intrinsic) luminosity comparable to the X-ray luminosity.
As a result, sources with $E_{\gamma,iso}\sim 10^{50-51}$ erg are not excluded
as possible TeV emitters, even though their detection is possible only for
relatively low redshift. This reduces the available volume, and hence the
detection rate of similar events. In any case, this is relevant also for short
GRBs which are less energetic than long ones, with typical isotropic energies
falling within the $\sim 10^{49-52}$ erg range Berger (2014).

###  5.4 X-ray lightcurves

The comparison between TeV and X-ray light-curves suggests an intimate
connection between the emission in these two bands, both in terms of emitted
energy and luminosity decay rate. In Figure 29 the XRT afterglow light-curves
(luminosity versus rest-frame time) in the $0.3-10$ keV energy range are
compared with the VHE light-curves (integrated over different energy ranges,
depending on the detection window, see Table 4). Different colors refer to the
six different GRBs. The VHE luminosity is shown with empty circles.

Considering the X-ray luminosity, the GRB sample can be divided into two
groups: GRB 190114C, GRB 180720B and GRB 201216C display large and clustered
X-ray luminosity (at $t\sim 10^{4}$ s their luminosity is around 1-5 $\sim
10^{47}$ erg s-1) and their light curves almost overlap for the entire
afterglow phase. The other three GRBs (GRB 190829A, GRB 201015A and GRB
160821B) are much fainter in terms of X-ray luminosity (at least two order of
magnitude at $t\sim 10^{4}$ s). This is consistent with the fact that they
also have a smaller $E_{\gamma,iso}$. The correlation between X-ray afterglow
luminosity and prompt $E_{\gamma,iso}$ is found in the bulk of the long GRB
population and these GRBs make no exception.

Observations in the VHE band (empty circles in Figure 29) reveal that the VHE
luminosities observed in the afterglow phase are in general smaller but
comparable to the simultaneous X-ray luminosity, implying that almost an equal
amount of energy is emitted in the two energy bands. Any theory aimed at
explaining the origin of the TeV radiation should explain the origin of these
similarity. Concerning the decay rate, observations are still not conclusive.
The decay rate of the TeV emission is available only for two events. For GRB
190829A the temporal indices in X-ray and VHE are very similar, while for GRB
190114C the VHE clearly decays faster than the X-ray emission.

For GRB 190114C at $t\sim 380$ s the VHE luminosity $L_{VHE}$ is $\sim
1.5-2.5\times 10^{48}$ erg s-1 and the X-ray one $L_{X}$ is $\sim
0.6-1.0\times 10^{49}$ erg s-1. As a result, the power radiated in the VHE
band is about $\sim 25\%$ of the X-ray one. Similarly for GRB 190829A at
$t\sim 4.5$ hrs the VHE luminosity $L_{VHE}$ is $\sim 4.0-8.5\times 10^{44}$
erg s-1 which is around $\sim 15-20\%$ of the corresponding X-ray one ($L_{X}$
$\sim 2.0-5.0\times 10^{45}$ erg s-1). For GRB 180720B at $t\sim 2\times
10^{4}$ s the VHE luminosity $L_{VHE}$ is $\sim 9\times 10^{47}$ erg s-1 and
the X-ray one $L_{X}$ is $\sim 1.5-2.5\times 10^{49}$ erg s-1. In this case
the power radiated in the VHE band is around $\sim 35-60\%$ of the X-ray one.

Figure 29: X-ray and VHE luminosity versus rest-frame time for the six GRBs
detected in the TeV domain. The X-ray light-curve in the 0.3-10 keV energy
range is taken from the Swift Burst Analyzer webpage .

https://www.swift.ac.uk/burst_analyser/

###  5.5 The TeV contribution to the multi-wavelength modeling

Modeling of multi-wavelength afterglow data provide important insights
concerning the GRB afterglow physics. In particular, the VHE data were crucial
to investigate (i) the radiation mechanisms responsible for the production of
photons between 10-100 GeV already detected by LAT; (ii) the environmental
conditions at the GRB site; (iii) the free parameters which describe the shock
micro-physics, and in particular the self-generated magnetic field.

The modelings proposed so far in literature to explain the VHE component have
considered two different radiation processes at the origin of the TeV
emission: SSC and synchrotron. In the first case, the VHE emission is
interpreted as a distinct spectral component from the synchrotron radiation
dominating from radio to $\sim$ GeV energies, which provides the seed photons
that are upscattered at higher energies by the same electron population. In
the second scenario, the VHE emission is seen as the extension of the
synchrotron spectrum up to TeV energies.

In principle, a simultaneous SED covering the X-ray, HE and VHE range should
be sufficient to discriminate among these two different possibilities. An
hardening of the spectrum from GeV to TeV energies should be the smoking gun
for the presence of a distinct component. In reality, the uncertainties in the
spectral slope at VHE (caused by the uncertainty on the EBL and on the narrow
energy range of TeV detection) can make the distinction hard to perform. In
this case, LAT observations are of paramount importance to reveal the presence
of a dip in the SED, which would also prove the need to invoke a different
origin for the VHE emission. This seems to be the case for GRB 190114C, for
which at least in one SED the LAT flux strongly suggests a dip in the GeV flux
and hence the presence of the characteristic double bump observed for a
synchrotron-SSC emission (Figure 20). For GRB 190829A, LAT provides only an
upper limit which is not constraining for modeling the shape of the SED
(Figure 24). In this GRB an interpretation of the whole SED in terms of
synchrotron radiation can not be excluded, although a modeling as SSC
radiation has been proved to be successful Salafia et al. (2021) (Figure 25).
For the other events detected at VHE, either the data do not allow to build a
proper SED with simultaneous multi-wavelength observations, or they are not
yet public. Despite this, the SSC emission seems to be the most viable
mechanism able to explain the TeV data. A firm conclusion on the responsible
radiation mechanism has not been reached yet and future detections will be
crucial for deeper investigations.

Assuming one of the two scenarios, TeV data coupled with broad band
observations at lower energies can be exploited to give additional information
on the details of the afterglow external forward shock scenario.

Concerning the shock micro-physics, several modeling have suggested the
possibility that the fraction of electrons accelerated in non-thermal
distribution, $\xi_{e}$, is different from the standard value of 1 which is
usually assumed. In a few GRBs modeling, namely for GRB 190114C Asano et al.
(2020) and GRB 190829A Salafia et al. (2021); Sato et al. (2021) the
introduction of a $\xi_{e}<1$ was essential in order to fit consistently the
observational data. In particular in Salafia et al. (2021) the requirement for
a low value of $\xi_{e}\lesssim 6.5\times 10^{-2}$ was required to provide
acceptable fit of the data. The other modeling assume a greater value of
$\xi_{e}$, around $\sim 0.3$. Further detections will be exploited in order to
verify if such indication could be present also in other events.

Some considerations can also be drawn for the equipartition parameters
$\epsilon_{e}$ and $\epsilon_{B}$. These values, especially the former one,
are usually well unconstrained and can span several order of magnitudes. The
TeV modeling described so far suggest that around $\sim 10\%$ of the energy is
given to the electrons, while a lower value (from $10^{-5}$ to $10^{-3}$) is
given to the magnetic field. Larger values of $\epsilon_{B}$ such as 0.1-0.01,
which are considered in external shock scenario, are excluded. Moreover, some
results also can be interpreted as an indication of an evolution in time of
these parameters. In Figure 22 the modeling of the broad band light curves of
GRB 190114C is shown. Two different modeling are presented: one optimized for
the early time X-ray, HE and VHE observations (solid line) and one optimized
for the late time lower energy bands (dotted line). This is due to the fact
that the model which reproduce the early time data over-predict the late time
optical and radio observations. This result point towards the possibility that
some of the fixed parameters of the afterglow theory (e.g. the electron and
magnetic field equipartition parameters) may evolve in time. A further clue of
the presence of time-dependent shock micro-physics parameters is derived from
the low frequency multi wavelength modeling of GRB 190114C presented in Misra
et al. (2021). In order to model the optical and the radio data it is required
that the micro-physical parameters evolve with time as $\epsilon_{e}\propto
t^{-0.4}$ and $\epsilon_{B}\propto t^{0.1}$ in the ISM case and
$\epsilon_{B}\propto t^{0.76}$ for the stellar wind scenario.

An issue that still is not solved by TeV observations is the discrimination
between constant and wind-like profile for the ambient density. It is expected
that long GRBs occur in wind-like environments. Nevertheless, at the current
stage there seems to be no preference between such environment and a constant
ISM one, which is able to well reproduce the observational data. Therefore
conclusive answers on the topic cannot be drawn yet.

In conclusion, the current population of GRBs at VHE already show quite broad
properties, spanning more than three order of magnitude in $E_{\gamma,iso}$
and more than two order of magnitude in terms of afterglow luminosity and
ranging in redshift between 0.079-1.1. The afterglow X-ray and VHE emission
have comparable fluxes and decay slopes. The afterglow emitted power in the
VHE band seems to constitute from $15\%$ up to $60\%$ of the X-ray one. Data
modeling suggest that the responsible VHE radiation mechanism is the SSC
emission although different mechanisms (e.g. synchrotron radiation, EIC)
cannot be completely excluded and a conclusive answer cannot be given yet.
Multi-wavelength modeling show no preferences concerning the GRB environments
between ISM or wind-like scenario and indicate that shock micro-physics
parameters which seems to be able to reproduce VHE emission are
$\varepsilon_{e}\sim 0.1$ and $\varepsilon_{B}\sim 10^{-5}-10^{-3}$. Such
features can be an indication of the universality of TeV emission in GRBs. It
is then expected that a larger sample of GRBs than the current one will be
detected in the VHE band, including also short GRBs for which, at the current
stage, there are no confirmed detection except for the hint of excesses seen
for GRB 160821B.

##  6 Conclusions and future prospects

The recent discoveries performed by current generation of Cherenkov telescopes
in the VHE band have opened a new observational spectral window on GRBs. The
presence of a TeV afterglow component has been unequivocally proven and the
studies on the currently available sample have shown the potential that such
detections have in probing several long-standing open questions in the GRB
field. These first studies have focused on the identification of the
responsible radiation mechanism, which is the first issue to address, and the
comparison of the energetics, luminosity, and temporal behaviour of the VHE
component with respect to emission at lower frequencies. Modeling of multi-
wavelength data covering from radio up to TeV energies were performed giving
interesting insights on the shock micro-physics conditions.

Limitations to the robust use of VHE data for afterglow modeling are imposed
by the severe modification of the intrinsic spectrum cased by the energy-
dependent flux-attenuation induced by EBL. GRBs with redshift $z>0.4$, four
out of six in the current VHE sample, are strongly affected by EBL absorption
starting from hundreds of GeV. This implies large uncertainties on the shape
and photon index of the intrinsic VHE spectrum. As a result, firm conclusions
on the origin and spectral regime of the TeV component cannot be drawn yet.
The low-energy extension of the range of sensitivity of IACTs is then
fundamental for reaching a larger rate of detections and a more robust
determination of the spectral index of the TeV component.

Figure 30: CTAO differential sensitivity(defined as the minimum flux needed to
obtain a 5-standard-deviation detection of a point-like source) for 50 h of
observations with the Northern and Southern array compared to the sensitivity
of several other Cherenkov telescopes and with Fermi-LAT (1 year).

https://www.cta-observatory.org/science/ctao-performance/ A full comprehension
and exploitation of TeV data is expected to be reached thanks to the next
generation of Cherenkov telescopes. The Cherenkov Telescope Array (CTA) will
be a huge step forward for the detection of GRBs in the VHE band. The major
upgrades with respect to the current generation of Cherenkov telescopes that
will impact GRB observations are: i) a lower energy threshold ($\lesssim 30$
GeV), ii) a larger effective area at multi-GeV energies ($\sim 10^{4}$ times
larger than Fermi-LAT at 30 GeV) and iii) a rapid slewing capability (180
degrees azimuthal rotation in 20 seconds). Moreover, its planned mixed-size
array of large, medium and small size telescopes (called LST, MST and SST,
respectively) situated at two sites in the northern and southern hemispheres
will provide a full sky coverage from few tens of GeV up to hundreds of TeV.
CTA will have a much better sensitivity and a broader energy range with
respect to current ground-based facilities. A comparison is shown in Figure
30. At the present stage, the first prototype of the LSTs has been built and
is operative under commissioning phasehttps://www.lst1.iac.es/ in the northern
site at the Roque de los Muchachos Observatory in La Palma. Despite these
performance improvements, the expected CTA detection rates of GRBs will be
anyway influenced by the relatively low duty cycle affecting IACTs and by the
synergies with other instruments. Indeed, Cherenkov telescopes repointing
relies on external triggers coming from space satellites. Assuming that
currently operating space telescopes will be still operative, GRB alerts will
be mostly provided by Swift-BAT and partially by the Fermi-GBM, and in the
future by the French-Chinese mission Space-based multi-band astronomical
Variable Objects Monitor (SVOM Wei et al. (2016)).

BAT observes around $92$ GRBs per year with a typical localization error of a
few arcmin Lien et al. (2016). The good localisation (later refined by XRT to
a few arcsec) is fundamental for Cherenkov telescopes, given their limited
field of view (e.g., about $4^{\circ}$ for the LSTs and $7^{\circ}$ for the
MSTs). The GBM provides a much higher number of alerts, around 250 per year
but with a larger localization error, from $1-3^{\circ}$ up to $10^{\circ}$,
which makes follow-up with IACTs very challenging. In case of such a large
localization errors, CTA can exploit the so-called divergent mode for
observations, which is currently under study Donini et al. (2019). In this
pointing strategy, each telescope points to a position in the sky that is
slightly offset to extend the field of view. Concerning future instruments,
SVOM is expected to provide Swift-like alerts at a rate of $\sim 60-80$
GRBs/yr with a localization error $<1^{\circ}$, including also 10 GRBs/yr with
redshift $z<1$.

Available estimates of CTA detection rate of GRBs are reported in Inoue et al.
(2013). These studies were performed before the discovery of TeV emission.
They are based on Swift-like alerts (triggered by Swift-BAT or SVOM) and
Fermi-GBM alerts. The predicted detection rate is around a few GRBs per year,
depending on the energy threshold of the observation and on the observation
delay Inoue et al. (2013). An updated study that considers current knowledge
of TeV emission in the afterglow of GRBs is in progress Grazia Bernardini et
al. (2019).

Despite for decades GRBs’ hunting by Cherenkov telescopes has been primarily
focused on reaching low energy thresholds in order to explore the multi-GeV
band, these first detections have shown that photons above TeV energies can be
produced in GRBs and can be detected. This is mostly valid only for nearby
events of redshift below $0.1-0.2$ where EBL attenuation is not too severe.
The exploration of the GRB emission component above 1 TeV can be of potential
interest for SSTs and for the ASTRI Mini-Array. The ASTRI Mini-Array,
currently under construction, will be an array of nine imaging atmospheric
dual-mirror Cherenkov telescopes at the Teide Observatory site, expected to
deliver the first scientific results in 2023. After the detection of GRB
190114C, the capabilities of the ASTRI Mini-Array in detecting and performing
spectral studies of an event similar to the MAGIC GRB have been explored
Stamerra et al. (2021). GRB 190114C has been taken as a template to simulate
possible GRB emission from few seconds to hours, and has been extrapolated to
10 TeV on the bases of model predictions.

Figure 31: Light-curve of GRB 190114C at 1 TeV (dotted purple curve) compared
to the sensitivity of the ASTRI Mini-Array. The yellow dashed and green dot-
dashed curves show how it would appear GRB 190114C rescaled at redshift
$z=0.25$ and $z=0.078$, respectively. From Stamerra et al. (2021).

The results show that the instrument will be able to detect afterglow TeV
emission from an event like GRB 190114C up to $\sim 200$ s (see the comparison
between the GRB observed flux at 1 TeV and the differential ASTRI Mini-Array
sensitivity in Figure 31). By moving the GRB at smaller redshift (down to
$z=0.078$, the redshift of the TeV GRB 190829A), the time for which the GRB is
detectable increases up to $\sim 10^{5}$ (however, the light-curve in this
case should be re-scaled by the lower energetics of nearby events). Nearby
GRBs are then potential target of interests for the ASTRI Mini-Array. These
are certainly rare events, but their detection will provide a wealth of
information, with spectra that can be characterised up to several TeV Stamerra
et al. (2021).

In conclusion, after decades of huge efforts, current ground-based VHE
facilities have started a new era in the comprehension and study of GRB
physics. Their breakthrough detections allow unprecedented studies. As
discussed in this review, many open questions in afterglow physics can largely
benefit from the inclusion of TeV data. The first detections are providing
glimpses of such a huge potential. Luckily, we are at the dawn of the VHE era
thanks to the upcoming CTA observatory, which will assure major upgrades in
sensitivity, energy range, temporal resolution, sky coverage. Future
observations, if complemented by simultaneous observations in X-rays and at
$\sim$ GeV energies, will play a paramount role to improve our knowledge on
the physics of GRB during the afterglow phase and hopefully also in the prompt
phase. In particular, the afterglow SSC one-zone model will be tested to
understand if it can grasp the main properties of the VHE emission or if a
revision of our comprehension on the particle acceleration processes, shock
micro-physics and radiation mechanisms is needed.

All authors, D.M. and L.N., have contributed to writing. All authors have read
and agreed to the published version of the manuscript. This research received
no external funding This work made use of data supplied by the UK Swift
Science Data Centre at the University of Leicester
(https://www.swift.ac.uk/xrt_curves/) The authors declare no conflict of
interest.

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