#  A photohadronic interpretation of the H.E.S.S. afterglow observations from
GRB 221009A

Sarira Sahu<EMAIL_ADDRESS>Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México,
Circuito Exterior S/N, C.U., A. Postal 70-543, CDMX 04510, México.  B. Medina-
Carrillo<EMAIL_ADDRESS>Departamento de Física Aplicada, Centro
de Investigación y de Estudios Avanzados del IPN,
Unidad Mérida. A. Postal 73, Cordemex, Mérida, Yucatán 97310, México.  D. I.
Páez-Sánchez<EMAIL_ADDRESS>Facultad de Ciencias, Universidad Nacional
Autónoma de México,
Circuito Exterior, C.U., A. Postal 70-543, 04510 CDMX, México.  G. Sánchez-
Colón<EMAIL_ADDRESS>Departamento de Física Aplicada, Centro de
Investigación y de Estudios Avanzados del IPN,
Unidad Mérida. A. Postal 73, Cordemex, Mérida, Yucatán 97310, México.  Subhash
Rajpoot<EMAIL_ADDRESS>Department of Physics and Astronomy,
California State University,
1250 Bellflower Boulevard, Long Beach, CA 90840, USA.

###### Abstract

The High Energy Stereoscopic System (H.E.S.S.) started observing the extremely
powerful long-duration gamma-ray burst, GRB 221009A, after 53 hours of the
triggering event. The H.E.S.S. collaboration carried out observations on the
11, 12, and 17, of October 2022 under poor atmospheric conditions, without
detecting significant very high-energy photons from the source and computed
the upper limits of the fluxes for the different nights. We study these flux
upper limits by using the photohadronic model and show that the interaction of
high-energy protons with the synchrotron seed photons in the forward shock
region of the GRB jet exhibits behavior compatible with the upper limits
computed by the H.E.S.S. collaboration.

Particle astrophysics (96), Blazars (164), Gamma-ray bursts (629),
Relativistic jets (1390)

mainReferences appendixReferences (Appendix)

##  1 Introduction

The long-duration gamma-ray burst, GRB 221009A, was first observed by the
Gamma-ray Burst Monitor (Meegan et al., 2009) onboard the Fermi Gamma-ray
Space Telescope on October 9, 2022 (Lesage et al., 2022; Veres et al., 2022).
The prompt emission was also observed by a fleet of space observatories (Gotz
et al., 2022; Kozyrev et al., 2022; Lapshov et al., 2022; Liu et al., 2022;
Ursi et al., 2022; Xiao et al., 2022; Frederiks et al., 2022; Dichiara et al.,
2022; Krimm et al., 2022; Piano et al., 2022; Mitchell et al., 2022; An et
al., 2023; Ripa et al., 2023; Williams et al., 2023). This is also the first
James Webb Space Telescope (JWST) observation of a GRB afterglow which
suggests that the extreme properties of GRB 22109A are not directly related to
its galactic environment (Levan et al., 2023). The redshift of the GRB is
measured to be $z\simeq 0.151$, corresponding to a luminosity distance of
$\sim 750$ Mpc (de Ugarte Postigo et al., 2022; Castro-Tirado et al., 2022).
This is the brightest GRB ever detected with the highest isotropic equivalent
total energy. For this reason it is refereed to as the brightest of all time
(BOAT) event (Burns et al., 2023). Observation of such a titanic GRB at a
closer distance is a very rare event (Williams et al., 2023; Malesani et al.,
2023). Estimates show that such rare events occur, on average, once every
10,000 years (Burns et al., 2023).

The Large High Altitude Air Shower Observatory (LHAASO) with its two
detectors, water Cherenkov detector array (WCDA) and larger air shower
kilometer square area (KM2A) detector (Cao et al., 2022), observed several
thousands very high-energy (VHE, $>100$ GeV) photons within $T_{0}+2000$ s in
the 500 GeV to 18 TeV energy range, where $T_{0}$ is the trigger time of GRB
221009A. Again, these are the most energetic photons ever observed from a GRB
(Huang et al., 2022). The propagating VHE photons interact with the
extragalactic background light (EBL) to produce $e^{+}e^{-}$ pairs
($\gamma\gamma\rightarrow e^{+}e^{-}$) (Stecker et al., 1992; Hauser & Dwek,
2001), thus attenuating the energy-dependent VHE photon flux reaching the
Earth. Several well known EBL models have been developed to study the
attenuation at different redshifts and photon energies (Franceschini et al.,
2008; Finke et al., 2010; Domínguez et al., 2011; Gilmore et al., 2012;
Saldana-Lopez et al., 2021). Particularly, the $\sim 18$ TeV photons from the
GRB 221009A should be attenuated by a factor of $\sim 10^{-8}$, making them
almost impossible to observe on Earth. In spite of this, observations of such
VHE photons from the GRB 221009A by LHAASO question the reliability of the EBL
models used in the very high-energy regime. To overcome this problem,
nonstandard physics explanations are proposed to explain the observation of
$\sim 18$ TeV photons (Baktash et al., 2022; Zhu & Ma, 2023; Galanti et al.,
2022; Troitsky, 2022; Brdar & Li, 2023; Nakagawa et al., 2023; Finke &
Razzaque, 2023; Li & Ma, 2023). Recent publication from the LHAASO
collaboration has shown that GRB 221009A was observed during the prompt and
the early afterglow emission phases in the VHE band by the WCDA. Photons of
energy $>0.2$ TeV were observed a few minutes after the triggering event. One
way to understand the rise and fall of the VHE spectra at different epochs is
entertained by using the leptonic scenario (LHAASO Collaboration et al.,
2023).

The H.E.S.S. collaboration started observing the GRB 221009A with its 12 m
telescopes after 53 hours of the triggering event when the observing condition
were favourable (Aharonian et al., 2023). On October 11, 2022, the H.E.S.S.
observation was for 32 minutes, followed by a second run of 32 minutes and
continued observing the source in the subsequent nights. Due to poor
atmospheric conditions and high aerosol content in the atmosphere (Formenti et
al., 2019), the only nights it could observe the source were on October 11,
12, and 17, of 2022 (Nights 3, 4, and 9, after the triggering event,
respectively). Additional observations undertaken on other nights were
excluded from the analysis due to the worsening of the atmospheric conditions
(Aharonian et al., 2023). Unfortunately, H.E.S.S. did not detect significant
emission of VHE gamma-rays from the source neither on the individual nights
nor in the combined data. Instead, the collaboration computed the upper limits
to constrain the VHE emission from GRB 221009A. In the energy range
$0.65\,\mathrm{TeV}\leq E_{\gamma}\leq 10\,\mathrm{TeV}$, the computed
integrated flux upper limit at 95% confidence level (CL) for Night 3 is
$4.06\times 10^{-11}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ and for all nights
combined is $9.7\times 10^{-12}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ (Aharonian et
al., 2023).

The relativistic jets of blazars and GRBs are pointing towards us and the
emission mechanisms in these sources show many similarities, despite large
differences in their masses, bulk Lorentz factors, and magnetic fields (Wang &
Wei, 2011; Nemmen et al., 2012; Wu et al., 2016).The relativistic jets in AGNs
and GRBs have many characteristics in common and have a similar energy
dissipation efficiency (Nemmen et al., 2012). So, the particle acceleration
mechanisms and the VHE emission mechanisms used in the context of blazars are
also used to study the VHE emissions in the afterglow phases of GRBs. We have
studied the multi-TeV flaring events from several blazars using the
photohadronic model. Recently, we have applied the photohadronic model
successfully to explain the VHE emission from the GRB 190114C, GRB 180720B and
190829A (Sahu & López Fortín, 2020; Sahu et al., 2022). Also, the observation
of $\sim 18$ TeV photons from GRB 221009A are studied using the photohadronic
model (Sahu et al., 2023).

Our aim here is to provide an alternative interpretation to the upper limits
of the VHE spectrum as observed by H.E.S.S. We do this in the context of the
photohadronic model and also analyze whether the seed photons in the
synchrotron regime or in the synchrotron self-Compton (SSC) regime can be
responsible for the results of these VHE afterglow observations.

The plan of the paper is as follows. In Section 2, we review the photohadronic
model and its application to explain the VHE flaring events from high-energy
peaked BL Lac objects (HBLs). In Section 3, the common features of blazars and
GRBs are discussed. Finally, we discuss the results and conclusions in Section
4.

##  2 The Photohadronic Model

Blazars belong to a subclass of active galactic nuclei (AGN) (Urry & Padovani,
1995; Romero et al., 2017). They have non-thermal spectra produced by the
relativistic jet oriented very close to the observer’s line of sight (Urry &
Padovani, 1995; Acciari et al., 2011). Rapid variability is a common feature
in their entire electromagnetic spectrum. The spectral energy distributions
(SEDs) of these blazars have a double peak structure (Abdo et al., 2010). The
low-energy peak is due to the synchrotron radiation of low-energy electrons in
the jet’s magnetic field. The second peak is from the Compton scattering off
the high-energy electrons with the self-produced synchrotron photons
(synchrotron self Compton, i.e., SSC) (Maraschi et al., 1992; Dermer &
Schlickeiser, 1993; Sikora et al., 1994; Murase et al., 2012; Gao et al.,
2013). To explain the VHE events from an HBL, a double jet structure scenario
is assumed (Sahu, 2019; Sahu et al., 2019). In this scenario, the inner jet is
smaller with a comoving radius $R^{\prime}_{f}$ (the notation ′ implies
quantity in comoving frame) and is enclosed by a larger cone of size
$R^{\prime}_{b}$ ($R^{\prime}_{f}<R^{\prime}_{b}$). Both the jets are moving
with a bulk Lorentz factor $\Gamma$ and a Doppler factor ${\cal D}$
(Ghisellini et al., 1998; Krawczynski et al., 2004). For HBLs,
$\Gamma\simeq{\cal D}$. The photon density in the outer jet region is
$n^{\prime}_{\gamma}$, which is estimated from the observed flux. On the other
hand, the photon density in the inner jet region $n^{\prime}_{\gamma,f}$ is
unknown. Adiabatic expansion of the inner jet to the outer jet decreases the
photon density in the inner region. To connect the inner and the outer jet
regions, we proposed a scaling behavior of the photon densities as (Sahu et
al., 2016)

$\frac{n^{\prime}_{\gamma,f}(\epsilon_{\gamma_{1}})}{n^{\prime}_{\gamma,f}(\epsilon_{\gamma_{2}})}\simeq\frac{n^{\prime}_{\gamma}(\epsilon_{\gamma_{1}})}{n^{\prime}_{\gamma}(\epsilon_{\gamma_{2}})},$
(1)

where $\epsilon_{\gamma}$ is the seed photon energy. In this equation, the
left hand side is unknown and the right hand side is known. We use the above
relation to express the unknown photon density in the inner region in terms of
the known photon density in the outer region. In the inner jet region, the
Fermi accelerated protons have a power-law spectrum given by

$\frac{dN}{dE_{p}}\propto E^{-\alpha}_{p},$ (2)

where $E_{p}$ is the proton energy and $\alpha\geq 2$ is the spectral index
for proton (Dermer & Schlickeiser, 1993). The value of $\alpha$ is different
for non-relativistic shocks, highly relativistic shocks and oblique
relativistic shocks (Keshet & Waxman, 2005; Summerlin & Baring, 2012). The
interaction of high-energy protons in the inner jet region produces the
$\Delta$-resonance ($p\gamma\rightarrow\Delta^{+}$), and subsequently, the
decay of the $\Delta$-resonance to $\gamma$-rays and neutrinos takes place by
the decay of the intermediate $\pi^{0}$ and $\pi^{+}$ respectively (Sahu et
al., 2012). Here we are concerned with the observation of $\gamma$-rays from
the $\pi^{0}$ decay. The kinematical conditions for the $\Delta^{+}$-resonance
process give

$E_{\gamma}\epsilon_{\gamma}\simeq 0.032\frac{{\cal
D}^{2}}{(1+z)^{2}}\,\mathrm{GeV}^{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}.}$
(3)

In the above process, the VHE photon carries about 10% of the proton energy
($E_{p}\simeq 10E_{\gamma}$).

The interaction of the VHE photons from extragalactic sources with the EBL
produces $e^{+}e^{-}$ pairs (Stecker et al., 1992; Ackermann et al., 2012;
Padovani et al., 2017). Thus, the VHE photon flux is attenuated by a factor
$e^{-\tau_{\gamma\gamma}}$, where the optical depth $\tau_{\gamma\gamma}$
depends on $E_{\gamma}$ and $z$. In the photohadronic model, the observed VHE
$\gamma$-ray flux is determined to be

$F_{\gamma}(E_{\gamma})=F_{0}\left(\frac{E_{\gamma}}{\rm
TeV}\right)^{-\delta+3}\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)}=F_{in}(E_{\gamma})\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)},$
(4)

where the normalization factor $F_{0}$ can be fixed from the observed
spectrum. The photon spectral index, $\delta=\alpha+\beta$, is the free
parameter in the model (Sahu, 2019; Sahu et al., 2019). $F_{in}(E_{\gamma})$
is the intrinsic flux and $\beta$ is the spectral index of the background seed
photons. It is further observed that for HBLs the seed photon flux is also a
power-law $\Phi_{\gamma}\propto\epsilon^{\beta}_{\gamma}\propto
E^{-\beta}_{\gamma}$. The sign of $\beta$ decides whether the seed photon
background is in the synchrotron regime ($\beta<0$) or in the SSC regime
($\beta>0$). By fitting the observed VHE spectrum, the value of $\delta$ is
fixed. For HBLs, we have shown that the value of $\delta$ is always in the
range $2.5\leq\delta\leq 3.0$ (Sahu et al., 2019). This makes the $\beta$
value to be positive.

##  3 Common features of Blazars and GRBs

The emission mechanisms in blazars and GRBs have many common features despite
enormous differences in their masses, bulk Lorentz factors, duration of
emission and other factors (Urry & Padovani, 1995; Wu et al., 2011; Wang &
Wei, 2011; Nemmen et al., 2012; Gehrels & Razzaque, 2013; Wu et al., 2016;
Zhang, 2018). In the past, we have studied very successfully the VHE gamma-ray
flaring mechanism in high-energy blazars (HBLs) using the photohadronic model
(Sahu, 2019; Sahu et al., 2019). Also, recently, the same photohadronic model
was employed successfully to explain the VHE gamma-ray spectra of the GRB
180720B, GRB 190114C and GRB 190829A (Sahu & López Fortín, 2020; Sahu et al.,
2022). Furthermore, the same model was employed to explain the observations of
$E_{\gamma}\gtrsim 10$ TeV photons from the GRB 221009A (Sahu et al., 2023).
Due to the extreme nature of the GRB 22109A, it is assumed that during the
afterglow epoch, the intrinsic photon flux from the source might have
increased by several orders of magnitude, which might help to compensate the
depletion from the $e^{+}e^{-}$ pair production of the VHE photons with the
EBL. Previously, it is shown that for GRBs, the $\beta$ value can be either
positive or negative (Sahu & López Fortín, 2020). The seed photons in the
synchrotron regime corresponds to $\beta<0$ (Sahu & López Fortín, 2020) and
$\beta>0$ corresponds to seed photons in the SSC regime (Sahu & López Fortín,
2020; Sahu et al., 2022). However, for HBLs $\beta>0$ always and
$2.5\leq\delta\leq 3.0$ (Sahu, 2019). In Sahu et al. (2023), it is
demonstrated that the $E_{\gamma}\gtrsim 10$ TeV photons from the GRB 221009A
can be explained in the context of the photohadronic process and
$1.2\lesssim\delta\lesssim 1.7$ is preferable. This range of $\delta$ implies
that the spectral index of the background seed photon flux is negative and the
seed photons are in the synchrotron regime in the forward shock region of the
GRB jet.

Figure 1:  In the context of the photohadronic model, using different values of the spectral index $\delta$ and the normalization factor $F_{0}$ (in units of $10^{-11}\mathrm{erg\,cm^{-2}\,s^{-1}}$), we have plotted the calculated flux together with the H.E.S.S. upper limits on Night 3 (October 11, 2022). For every observed spectrum $F_{\gamma}(E_{\gamma})$ (continuous curve), we have shown the corresponding intrinsic spectrum $F_{in}(E_{\gamma})$ (dashed curve of the same color). Figure 2: In the context of the photohadronic model, using different values of the spectral index $\delta$ and the normalization factor $F_{0}$ (in units of $10^{-11}\mathrm{erg\,cm^{-2}\,s^{-1}}$), we have plotted the calculated flux together with the H.E.S.S. upper limits on All nights combined (Nights 3, 4 and 9). For every observed spectrum $F_{\gamma}(E_{\gamma})$ (continuous curve), we have shown the corresponding intrinsic spectrum $F_{in}(E_{\gamma})$ (dashed curve of the same color).  Table 1: For different values of $\delta$ we have calculated the normalization factor $F_{0}$ and the integrated flux $F^{int}_{\gamma}$ (in units of $10^{-11}\mathrm{erg\,cm^{-2}\,s^{-1}}$) in the energy range $0.65\,\mathrm{TeV}\leq E_{\gamma}\leq 10\,\mathrm{TeV}$ for Night 3 and for All nights combined. Corresponding to the integrated flux we have also estimated the luminosity $L_{\gamma}$ (in units of $10^{44}\,\mathrm{erg\,s^{-1}}$). $\delta$ | Interval | $F_{0}$ | $F^{int}_{\gamma}$ | $L_{\gamma}$
---|---|---|---|---
1.2 | Night 3 | 0.77 | 0.79 | 5.32
| All nights | 0.28 | 0.29 | 1.95
1.5 | Night 3 | 0.94 | 0.72 | 4.84
| All nights | 0.34 | 0.26 | 1.75
1.7 | Night 3 | 1.10 | 0.69 | 4.64
| All nights | 0.39 | 0.25 | 1.68
1.9 | Night 3 | 1.25 | 0.67 | 4.51
| All nights | 0.45 | 0.24 | 1.61
2.5 | Night 3 | 1.75 | 0.64 | 4.31
| All nights | 0.86 | 0.32 | 2.15

##  4 Results and Conclusions

In spite of poor atmospheric conditions and high levels of moonlight, the
H.E.S.S. collaboration estimated with 95% CL, upper limits to the intrinsic
fluxes on October 11 (Night 3) and the combined data set for the Nights of 3,
4, and 9. The striking feature of the estimated spectrum is that it increases
slowly, peaks around 5 TeV and then decreases exponentially. By incorporating
the EBL model of Franceschini et al. (2008) in the photohadronic model, we
plotted the upper limits for Night 3 and for the combined fluxes of all the
nights by using different values of $\delta$ and $F_{0}$. The results are
shown in Figure 1 and Figure 2, respectively. Here, our analysis is based on
the EBL model of Franceschini et al. (2008). It is observed that the EBL model
of Domínguez et al. (2011) has similar results. In Figure 1, it is observed
that for $1.2\lesssim\delta\lesssim 1.9$ the predicted spectrum is compatible
with the H.E.S.S. upper limits. The intrinsic flux in this model is given by
$F_{in}\propto E^{1.8-1.1}_{\gamma}$. Also, the estimated photohadronic model
curves have peaks around 5 TeV, consistent with the H.E.S.S. estimate.
However, for $\delta>1.9$ (e.g. $\delta=2.5$), the flux has a large value in
the low energy regime and decreases as the energy increases. Irrespective of
the upper limits, this trend can only be explained by taking
$1.2\lesssim\delta\lesssim 1.9$ and by adjusting the normalization factor
$F_{0}$. In this case, the flux slowly increases to a large value for
$E_{\gamma}\sim 5$ TeV and then decreases. However, for $\delta\gtrsim 2.0$
the flux slowly decreases from a large value and is thus unable to explain the
trend. From the above analysis, we conclude that large values of $\delta$
($\delta\gtrsim 2.0$) are not compatible with the H.E.S.S. results. We have
also shown the upper limits to the intrinsic fluxes corresponding to each
value of $\delta$ in the same figure. The same behavior is also observed for
the flux from the combined data set, as shown in Figure 2. In the context of
the photohadronic model, the VHE gamma-ray photons observed during the
afterglow (within $T_{0}$+2000 s) by LHAASO and the upper limits reported from
the 3rd to the 9th nights by H.E.S.S. are produced from the interactions of
the Fermi accelerated high-energy protons with the photons in the descending
part of the synchrotron spectrum in the forward shock region of the GRB jet.
It is similar to the VHE gamma-ray flux observed during the afterglow from GRB
180720B (Sahu & López Fortín, 2020). The GRB 221009A could produce multi-TeV
gamma-rays for more than a week after the prompt phase. This feature is unique
to GRB 221009A. In Table 1 we have shown the normalization constant $F_{0}$,
the integrated flux $F_{\gamma}^{int}$, and the luminosity $L_{\gamma}$ in the
energy range $0.65\,\mathrm{TeV}\leq E_{\gamma}\leq 10\,\mathrm{TeV}$ for the
observations on Night 3 and the combined nights for different values of the
spectral index $\delta$. The estimated $F_{\gamma}^{int}$ for each $\delta$ is
smaller than the upper limit of H.E.S.S. (Aharonian et al., 2023).

The VHE photons in the energy range $0.65\,\mathrm{TeV}\leq E_{\gamma}\leq
10\,\mathrm{TeV}$ correspond to protons in the energy range
$6.5\,\mathrm{TeV}\leq E_{p}\leq 100\,\mathrm{TeV}$. These protons are
accompanied by the electrons in the same energy range. Both the electrons and
the protons radiate synchrotron photons in the GRB jet (Mannheim et al., 1991;
Mücke & Protheroe, 2001; Mücke et al., 2003; Kelner & Aharonian, 2008). The
synchrotron photons radiated by the protons will be suppressed by a factor of
$m^{-4}_{p}$, where $m_{p}$ is the proton mass, relative to the synchrotron
photons from the electrons. If the jet has a magnetic field of $\sim 1$ G,
then the synchrotron photons from the electrons will be radiated in the energy
range $2.2\,\mathrm{MeV}\leq\epsilon_{\gamma}\leq 522\,\mathrm{MeV}$ which
falls in the lower part of the SSC spectrum. However, for smaller magnetic
fields, this range of $\epsilon_{\gamma}$ will be reduced.

If VHE photons were to interact with the seed photons in the GRB jet to
produce $e^{+}e^{-}$ pairs (Petropoulou & Mastichiadis, 2009), then each
lepton will have maximum energy of $E_{\gamma}/2$. Similarly, the $e^{+}$
produced from the $\pi^{+}$ decay will have energy in the range
$0.33\,\mathrm{TeV}\leq E_{e}\leq 5\,\mathrm{TeV}$. Thus, in the jet magnetic
field, the $e^{+}$ and $e^{-}$ will emit synchrotron photons in the energy
range $6\,\mathrm{keV}\leq\epsilon_{\gamma}\leq 1.3\,\mathrm{MeV}$.

The GeV-TeV photons produced from the $\pi^{0}$ decay will mostly encounter
the seed photons with energy $\epsilon_{\gamma}\geq 6$ keV as discussed above.
The $e^{+}e^{-}$ pair production cross section for $\epsilon_{\gamma}\geq 6$
keV is $\sigma_{\gamma\gamma}\lesssim 10^{-29}\,\mathrm{cm^{2}}$. After 53
hours of the triggering event of the GRB 221009A, the jet has a radius
$R^{\prime}_{b}\sim(10^{17}-10^{18})$ cm. The mean free path for the
$e^{+}e^{-}$ process is
$\lambda_{\gamma\gamma}=(n^{\prime}_{\gamma}\sigma_{\gamma\gamma})^{-1}$. For
$\lambda_{\gamma\gamma}\leq R^{\prime}_{b}$ we need the photon density in the
jet to satisfy $n^{\prime}_{\gamma}\geq 10^{11}\,\mathrm{cm^{-3}}$ and such
high photon density is difficult to achieve far away from the central engine.
Thus, the VHE photons will not be attenuated in the jet environment.

Another process, the Bethe–Heitler (BH) pair production process
$p\,\gamma\rightarrow$ $p\,e^{+}e^{-}$, can also compete with the
photohadronic process. The produced $e^{+}e^{-}$ pair can also emit
synchrotron photons (Petropoulou & Mastichiadis, 2015). The BH process may be
important for very high-energy protons and electrons, but here their
contribution is negligible in the present context (Petropoulou et al., 2015).
Thus, we conclude that the proton synchrotron process and the synchrotron
process from the electrons and positrons will not affect the propagation of
the VHE photons in the GRB jet environment.

We conclude that, in the context of the photohadronic scenario, the H.E.S.S.
upper limits on the VHE flux in the energy range $0.65\,\mathrm{TeV}\leq
E_{\gamma}\leq 10\,\mathrm{TeV}$ are compatible, provided the high-energy
protons in the GRB jet interact with the descending part of the synchroton
spectrum and not with the SSC photons. Future observations of more GRBs in VHE
gamma-rays are required to establish the validity of our claim.

We thank the referee for corrections/suggestions and constructive remarks that
have considerably improved the manuscript. The work of S.S. is partially
supported by DGAPA-UNAM (México) Projects No. IN103522. B. M-C and G. S-C
would like to thank CONACyT (México) for partial support. Partial support from
CSU-Long Beach is gratefully acknowledged. S.S. is thankful to J. D.
Mbarubucyeye of the H.E.S.S. collaboration for fruitful discussions.

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