Light scalar explanation for 18 TeV GRB 221009A

Recent astrophysical transient Swift J1913.1+1946 may be associated with the $\gamma$-ray burst GRB 221009A. The redshift of this event is $z\simeq 0.151$. Very high-energy $\gamma$-rays (up to 18 TeV) followed the transient and were observed by LHAASO, additionally Carpet-2 detected a photon-like air shower of 251 TeV. Photons of such high energy are expected to readily annihilate with the diffuse extragalactic background light (EBL) before reaching Earth. If the $\gamma$-ray identification and redshift measurements are correct, new physics could be necessary to explain these measurements. This letter provides the first CP-even scalar explanation of the most energetic 18 TeV event reported by LHAASO. In this minimal scenario, the light scalar singlet $S$ mixes with the Standard Model (SM) Higgs boson $h$. The highly boosted $S$ particles are produced in the GRB and then undergo the radiative decay di-photon $S\rightarrow \gamma\gamma$ while propagating to Earth. The resulting photons may thus be produced at a remote region without being nullified by the EBL. Hence, the usual exponential reduction of $\gamma$-rays is lifted due to an attenuation that is inverse in the optical depth, which becomes much larger due to the scalar carriers.


I. INTRODUCTION
Gamma-ray burst GRB 221009 A events over energies from 500 GeV extending up to 18 TeV have been recently detected. The events are reported to have occurred at a redshift of z ≈ 0.15 corresponding to a distance of d = 645 Mpc [1]. The initial detection was first reported by the Swift-Burst Alert Telescope [2], the Fermi satellite [3] and CARPET-2 [4]. The number of photons detected by LHAASO [5] in an interval of 2000 s was O(5000) events. However, observing such energetic photons is extremely unlikely since the flux is expected to be rapidly attenuated when propagating and interacting with extragalactic particles [6][7][8][9]. The unlikeliness for photons in the upper band of the 500 GeV-18 TeV range to survive in this manner while propagating to the Earth motivated several studies beyond the Standard Model (SM) as possible solutions. These beyond-the-SM models include explanations from sterile neutrinos [10][11][12][13], axion-like particles [14][15][16][17][18] and axion-photon conversion scenarios [19]. Even SM explanations have recently been discussed, where external inverse-Compton interactions with the cosmological radiation fields and proton synchrotron emissions could generate the ultra-energetic γ-rays [20][21][22][23].
To accommodate the aforementioned high energy photon observations, the S particles corresponding to this new field are produced in large numbers during the GRB through hadronic scattering and then undergo the radiative decay S → γγ while propagating to Earth. The resulting decay photons may therefore be produced at a region with less interstellar medium and avoid being nullified. Hence, the S particles provide an effective means for the survival of the photons to Earth. More explicitly, this corresponds to the usual exponential reduction of γ-rays being lifted due to an attenuation that is inverse in the optical depth.
In this work, we will use the 18 TeV GRB γ-ray flux observation and the number of the detected events in the 2000 second observation window to set new astrophysical bounds. These limits complement the existing bounds derived from stellar cooling and luminosity measurements of the Sun, white dwarfs, red giants and supernovae [58][59][60].
The outline of this paper is as follows. In Sec. II we introduce the simplified extension to the SM scalar sector and show the decay rates of the scalar S into various states. In Sec. III we compute the flux of γ-rays produced by the S-decay. Finally, we summarise and discuss the GRB bounds on the scalar mass and mixing in Sec. IV and conclude in Sec. V.

II. SCALAR DECAY
This section reviews the decay rates of the CP-even scalar S into γ-rays. Therefore, we first introduce the scalar interactions with the SM particles. At tree-level, the S interacts with leptons and pions via mixing with the SM Higgs. The interaction Lagrangian for these specific interactions is given by where A π is the effective coupling of S to pions [61,62]. sin θ is the S-mixing with the SM Higgs h and denotes leptons. 1 Hence, the scalar can decay into photons at one-loop level as well as electrons, muons and pions at tree-level from mixing with the Higgs boson. For S scalars, with masses below twice the electron mass (m S < 2m e ), the dominant channel is the decay to two photons (S → γγ). The decay rate for this process is given by [63], where α = e 2 /4π is the fine structure-constant and v = 246 GeV is the Higgs vacuum expectation value. When the S scalar masses are above twice the lepton mass threshold (m S 2m ), the scalar can decay into two oppositely charged leptons respectively. In this case, the decay rate (S → + − ) is given by [64], where m is either the electron or muon mass depending on the decay channel of interest. Here, we will only consider the scalar massive enough to decay into a pair of electrons e.

III. PHOTON FLUX
We now compute the γ-ray flux Φ γ from the S scalar decay. First, we introduce the γ-ray flux Φ 0 γ of GRB 221009A, which corresponds to the flux where all photons survive their interactions with the extragalactic medium on their way to the Earth, This is the so-called unattenuated flux. As discussed in Ref. [10], this is ascertained by extrapolating the flux measured by Fermi-LAT (GCN 32658) in [0.1,1] GeV to TeV scale [14]. We define the decay-production probabilities that a single particle S decays into two photons, and they reach the Earth. If the S scalar decay occurs at a distance interval of [x, x + dx], the decay-production probability is then given by [10], d is the distance at which the flux was produced. B γ is the branching ratio for the scalar decay into photons. The second exponential factor is the probability that the produced γ-rays survive their trajectory and is given in terms of the photon absorption length (or optical depth) τ = d/λ γ where λ γ is the mean-free-path of the photon. In this expression, the λ S is the mean free path (MFP) of the S scalar in terms of the S decay rate Γ S and energy of the scalar E S in the Earth's rest frame To compute the γ-ray flux produced from the S scalar decay, we integrate the expression given in Eq. (5) over x and multiply by the incoming S scalar flux at Earth Φ S from the GRB source, that would be expected if the scalar did not decay the factor of 2 is included to account for the two photons produced in the S-decay. Rewriting this expression in terms of the photon absorption length τ (E γ ) leads to  obstacles, such as supernova remnants [71]. The collision of the supernova ejecta with the high-energy jets could lead to the formation of an interaction region, in which particles can be accelerated and produce high-energy scalar emission [72]. Scalars can also be produced by heavy meson decays. With present information about the GRB, the exact fraction of energy that can be extracted from the GRB in exotic states is unknown. However we can make some conservative estimates and adjust them in them in the event of more refined measurements in the future. The amount of energy emitted in a supernova explosion is typically about 10 44 J ≈ 6.2 × 10 62 eV. In general, almost all of this energy is carried away by neutrinos. In the case of the GRB of interest here, the energy output is estimated at 2 × 10 54 erg = 1.2 × 10 66 eV [73]. Assuming that almost all the energy is lost and emitted as neutrinos, and using 0.3% of this energy as an upper bound for scalar emission with a rectangular energy distribution in the range [1,36] TeV corresponding to an average energy of E S = 18.5 TeV 2 , we get a scalar flux (assuming no decays) of Φ S 5 × 10 −23 eV −1 cm −2 s −1 at Earth over the 2000 s duration. It should be noted that this is very conservative since when considering supernova bounds, typically, the flux of the new particle is bounded by 10% of the neutrino luminosity, see, e.g. Refs. [58,67,70]. It 2 Note that the rectangular spectrum in the energy of the scalar flux at the GRB source between [1,36] TeV is chosen because this results in photon energies of Eγ E S /2 which lies in the [0.5, 18] TeV range LHAASO observations window. In the absence of more information about the GRB source energy profile, this is the simplest possible choice.
should be noted that E S = 2 E γ is fixed simply due to kinematics of two-body decays. Since the max energy for the GRB photons is observed at 18 TeV, the energy of the parent scalar has to be at least twice this energy. Other, more exotic beyond the SM scenarios can also be invoked to generate the scalar flux from the source, but we do not explore them in this work.
Finally, we can observe in Fig. 1 that the flux of γ-rays from the S scalar decay increases at high energies, approaching the unattenuated γ-ray flux. This indicates that the photons can survive interactions with the extragalactic background.

IV. BOUNDS ON THE SCALAR MASS-MIXING PARAMETER SPACE
The flux in the previous section provides us with a means to set bounds on the scalar mass-mixing parameter space. We require the number of photons produced by the scalar decay to be N S γ = 50, 500 and 5000 respectively. Here N S γ is obtained by integrating Eq. (8) over the photon energy E γ over the range of [0.5, 18] TeV as well as integrating over the time t from 0 s to 2000 s and the detector area of 1 km 2 [74,75].
We consider scalars with masses up to 2 MeV. Therefore, these scalars can only decay directly into photons and electrons. It is easy to find and extrapolate the relationship between sin θ and m S as a function of the number of events expected in 2000 s. This mixing-mass relationship providing upper bounds is found to be given by the relation This relationship also depends on the scalar flux Φ S and assumes an even distribution of Φ S over the E S range [1,36] TeV. In Fig. 2, we show the upper bounds for light scalar decaying into photons to explain GRB 221009 A in the scalar mixing-mass plane. This was computed numerically using the expression in Eq. (8) considering three benchmarks for the number of events N S γ = 50, 500 and 5000. We also show the stellar exclusions in Ref. [58] given by the supernova SN1987A using different numerical profiles Fischer 11.8M (blue), Fischer 18M (cyan) [78] and Nakazato 13M (magenta) [79]. We also show the combined collider searches (gray) [63,76,77]. Note that when there is an order of magnitude difference between the expected number of events, the difference between the bounds is a factor of a few. For the case of two orders of magnitude difference in N S γ , we find an order of magnitude difference between the limits. This is easily understood from the proportionality between the mixing and the number of events expected FIG. 2. Limits for light scalar decays into photons to explain GRB 221009 A in the scalar mass-mixing (mS, sin θ) plane requiring N S γ ≥ 50, 500 and 5000 number of events. The shaded regions correspond to supernova exclusions from Ref. [58] from various models such as Fischer 11M (blue), Fischer 18M (cyan) and Nakazato 13M (magenta). Additionally, we add complementary limits from SN1987A and collider searches [63,76,77].
scales like sin θ ∝ N S γ . Additionally, we check the MFP of the scalar at an average energy of E S = 18.5 TeV for the three different expected event boundaries shown in Fig. 2, these are on the order of 10 30 cm (50 events), 10 29 cm (500 events) and 4 × 10 26 -5 × 10 27 cm (5000 events) which is comparable to the estimated GRB source distance of 10 27 cm.
The bounds obtained from GRB 221009 A by requiring N S γ = 50, 500 and 5000 events complement those given by supernovae [58] and collider exclusions [63,76,77]. Note that considering regions with > 500 or > 50 events, there is no upper bound on mixing. However, there is a small band in Fig. 2 for > 5000 events because, in this region, λ S d and hence the photon flux is locally maximised. This can be understood as follows: At higher mixing, the scalars decay closer to the source, and more of the resulting photons get attenuated by the EBL. However, even if the scalars decay very quickly near the source, some minimum number of lower energy photons in the [0.5, 18] TeV range will still survive their journey to Earth, albeit with a smaller number of events, as shown. The lower bounds on mixing are because the scalars are too long-lived. On average, they may decay well beyond the Earth, reducing the observed photon flux and hence the event yield.
While supernova bounds excluded parameter space across the whole scalar mass range between mixing of the order of 10 −7 -10 −5 , the excluded area from GRB 221009 A intersects the excluded area from the supernovae limits, at masses between a few 10-10 3 keV, in the same mixing interval. However, GRB 221009 A excludes a large parameter of space above a mixing of 10 −5 . It is important to notice that the GRB 221009 A gives a better constraint than the supernova for a scalar with a mass of m S = 10 3 keV at a fixed mixing of sin θ 10 −8 .
The time delay ∆t in arrival (or time of flight difference) to travel a distance d between a massive particle of mass m S and a massless particle is given by the simple formula ∆t = We require that the time delay is smaller than the GRB duration of 2000s. We can see from this that the largest time delay is controlled by the higher mass scalars with lower total energy. To approximate the time delay, we consider the limiting mass for an average energy in the scalar spectrum of E S , which yields m S 4.5 × 10 3 keV, well outside our allowed region in Fig. 2 which is cut-off at 10 3 keV. Hence the allowed region is not affected by the time delay and complies easily with the 2000s requirement. If, however, additional information arises enabling the identification of the lowest photon energy in the range as low (or lower) than 0.5 TeV from S decays, this would strengthen the bound to lower masses, although such an identification is unlikely given the large background from conventional photons in this region. Conversely, if a very high energy photon is identified with a long time delay event, this would indicate higher m S . For massless particles in the final state produced by the decay of a massive boosted scalar, like in our scenario, the angular dispersion Θ in extremely relativistic limit in the lab frame is given in general by Θ = 2m S E S . Taking the maximum mass of interest as a limiting scenario and the average scalar energy, we get Θ 10 −7 . Hence, as long as the GRB jet opening angle exceeds this tiny number, there is no additional suppression of the γ-rays from S at the Earth.

V. CONCLUSION
In this work, we have considered a generic scalar CPeven S scalar corresponding to a CP-even singlet extension of the SM scalar sector as an explanation for GRB 221009 A. We consider a scenario wherein the S particles are produced in the GRB through hadronic scattering and then undergo the radiative loop-level decay S → γγ while propagating to Earth. If the photons are produced without being nullified by the extragalactic background, it could explain the anomalously high observed γ−ray emission at 18 TeV.
We calculated the flux of γ-rays from this process for different benchmarks of the expected number of events on Earth. We found that the γ-ray flux produced increases at higher photon energies and approaches the unattenuated γ-ray flux.
We have also computed bounds on the mixing-mass parameter space of the S scalar for different fractions of the observed LHAASO events. We found that at masses between a few10-10 3 keV, it overlaps with the excluded parameter space set by the SN1987A luminosity bounds. However, an upper bound exists in the parameter space of large mixing for all masses between 10 1 -10 3 keV that is not excluded by SN1987A or collider measurements. We also found that the strongest S constraint for GRB 221009 A is for a scalar with a mass of 10 3 keV and a mixing of 10 −8 for 50 events.