Internal Shock with a Background Magnetic Field for the Prompt Emission of Gamma-Ray Bursts—A Case Study of GRB 211211A

It is proposed that the synchrotron emission from an internal shock with a decaying shock-generated magnetic field can account for the prompt emission of gamma-ray bursts (GRBs). Generally, a jet from the central engine of a GRB is launched with a significant magnetization, and thus there would be a background magnetic field, rather than only the shock-generated magnetic field, in the emission region. In this paper, we study the synchrotron emission of internal shocks with both a decaying shock-generated magnetic field and a nondecaying background magnetic field. It is found that a shoulder with spectral index −1/2 appears in the low-energy regime of the radiation spectrum. The shoulder becomes dominant by increasing the ratio of the background magnetic field energy to the initial value of the shock-generated magnetic field energy f B. Correspondingly, a radiation spectrum with two bumps or a plateau around the peak of the ν F ν −ν spectrum may appear. Owing to the decay of the shock-generated magnetic field, the radiation spectral morphology in the high-energy regime is not a power-law function even though a power-law distribution of electrons is injected. We apply our model to GRB 211211A, of which the hard main emission is suggested to originate from the synchrotron emission. Compared with the spectral fitting results with a Band function and the synchrotron emission from the standard straightforward internal shocks, our model presents a perfect fitting to the observations. The fitting results show that f B is around 0.41–0.99 for the hard main emission of this burst.


Introduction
Gamma-ray bursts (GRBs) are the brightest explosions in the Universe and are characterized by high variability.Though much progress has been made in understanding the central engine and the afterglow radiation, the radiative mechanism of GRB prompt emission remains an open question.The observed radiation spectra of prompt emission have a distinct nonthermal component and are typically fitted by using a Band function, i.e., exponentially connected broken power laws (Band et al. 1993).There are some energy dissipation models, such as the fireball internal shock model (Rees & Meszaros 1994;Kobayashi et al. 1997), magnetic reconnection (Zhang & Kobayashi 2005), the dissipative photospheric model (Rees & Mészáros 2005;Giannios 2008;Beloborodov 2010;Lazzati & Begelman 2010;Veres et al. 2012), and an internal-collisioninduced magnetic reconnection and turbulence model (Zhang & Yan 2011).But no single model can properly explain most of the GRB prompt radiation.According to the nonthermal characteristics of the radiation spectrum, synchrotron radiation and inverse Compton radiation are widely discussed as promising radiation mechanisms that might be responsible for the prompt radiation (Tavani 1996;Lloyd & Petrosian 2000;Zhang & Mészáros 2002;Daigne et al. 2011;Zhang & Yan 2011).Since the typical parameter E p of the Band function is ∼250 keV, where E p is the peak photon energy of the νF ν −ν spectrum (Preece et al. 2000), synchrotron radiation appears to be a better explanation of the main mechanism for GRB prompt radiation than inverse Compton radiation.However, the photon spectral index α of the Band function is −1, while the theoretical spectrum of the synchrotron in the GRB scenario is −3/2 (Sari et al. 1998).This is the so-called fast cooling problem (Preece et al. 1998;Ghisellini et al. 2000;Kumar & McMahon 2008).Hence, some modified synchrotron radiation models have been proposed to solve this problem.These include anisotropic distributions of pitch angle (Lloyd & Petrosian 2000;Medvedev 2000), electrons cooling via inverse Compton scattering (Derishev et al. 2001;Nakar et al. 2009;Daigne et al. 2011), a magnetic field decaying with distance (Uhm & Zhang 2014), and coherently considering the adiabatic, synchrotron, and inverse Compton cooling mechanisms together with a decaying magnetic field (Geng et al. 2018).
In the widely discussed internal shock model, the energy dissipation of the GRB is caused by a fast jet shell colliding with a slow jet shell.The shock waves generated by these collisions can accelerate the electrons and produce a random magnetic field.In the internal shock model, the random magnetic field is generated by the electric current filaments generated from Weibel instability (Weibel 1959).Some numerical simulations show that the filaments only survive on a microscopic scale behind the shock front, and then they would interact with each other and undergo forced coalescence in the downstream region (Silva et al. 2003;Medvedev et al. 2005;Chang et al. 2008).Thus, the magnetic field strength generated by the shocks decays with time.This kind of decaying magnetic field in the GRB emission region with adiabatic cooling and inverse Compton cooling also leads to a harder synchrotron spectrum than the fast cooling spectrum, which can explain the Band spectrum (Pe 'er & Zhang 2006;Zhao et al. 2014), unlike the standard synchrotron emission model in an internal shock.Generally, the GRB central engine originally carries a strong large-scale magnetic field component.Several scenarios for the GRBs' central engine have been discussed in the literature.The leading type of these scenarios is a stellar-mass black hole surrounded by a hyperaccretion disk (Narayan et al. 1992(Narayan et al. , 2001;;Popham et al. 1999;Liu et al. 2015).It is generally believed that the magnetic field around a black hole is supported by the surrounding hyperaccretion disk.By equating the magnetic pressure on the black hole horizon to the ram pressure of the accretion flow at the inner radius r in , the magnetic field strength around the black hole can be roughly estimated as , where  M acc is the accretion rate of the accretion flow, M solar is the solar mass, and r in ∼ 10 6 cm is adopted.A millisecond magnetar has also been suggested as the central engine of GRBs (Usov 1992;Thompson 1994).A magnetar is a strongly magnetized neutron star with a surface dipolar magnetic field strength of the order of 10 14-16 G.In these two above scenarios, the strength of the magnetic field in the inner region of the corresponding launched jet can be of the order of 10 13-16 G.When the front of the jet reaches a large distance, the magnetic field energy from the central engine may be not fully dissipated.There may be a large-scale background magnetic field in the emission region.As the random magnetic field decays to be less than the background magnetic field, the latter would become dominant.Thus, the energy spectrum generated in this case may be different from that in the case without a background magnetic field.In this paper, we study the synchrotron radiation of the internal shock with both a decaying shock-generated random magnetic field and a nondecaying background magnetic field.
The paper is organized as follows.In Section 2, the emission of the internal shock is presented, where the internal shock with both a decaying shock-generated magnetic field and a nondecaying background field is our focus.In Section 3, we show the numerical results for the synchrotron radiation spectrum in internal shocks with both a decaying shockgenerated magnetic field and a nondecaying background field.In Section 4, we perform the spectral fitting on the hard main emission of GRB 211211A, of which the radiative emission is suggested to originate from the synchrotron emission.Spectral models of the Band function, the synchrotron emission from standard straightforward internal shocks (i.e., scenarios where the shock-generated magnetic field does not decay with time), and our model are all involved.The conclusions are presented in Section 5.

Emission of the Internal Shock
We focus our attention on the emission of the internal shock, which is formed during the collision of GRBs' jet shells with different velocities.At and near the shock front, the electrons are accelerated, and random magnetic fields are formed.While flowing downstream, electrons produce synchrotron emission within the magnetic field in situ and inverse Compton radiation.Hereafter, the superscript prime is used to denote quantities measured in the comoving frame of the downstream region, and the subscript " ...,0 " is used to denote the quantities that are associated with those generated at the time ¢ t 0 , where ¢ t 0 is the time measured in the rest frame of the downstream region since the collision of two jet shells.
Magnetic Field Prescription.The simulations reveal that the random magnetic field generated by the shock decays with time in a power-law form when it flows away from the shock front to the downstream region (Chang et al. 2008;Lemoine 2013;Lemoine et al. 2013).Then, the shock-generated random magnetic field can be described as T m sh,0 and g ¢ m being the minimum Lorentz factor of the shock-accelerated electrons (Zhao et al. 2014).The value of τ B = 0.01 is adopted in this paper, and thus the decay timescale of the shock-generated magnetic field is around 10 3 /ω p , where ω p is the proton plasma frequency.The decay timescale of the shock-generated magnetic field is found to be around w --10 10 2 3 p 1 in simulations (Chang et al. 2008;Keshet et al. 2009;Lemoine 2013;Lemoine et al. 2013) In general, the initial strength of the shock-generated random magnetic field ¢ B sh,0 is associated with the dissipated energy density of the shock e as where ε B is the fraction of the dissipated shock energy used to form the random magnetic field.With the dissipation efficiency ε dis , the dissipated energy e' of the shock can be described as , where L k is the kinetic energy of the jet, R dis is the dissipation location, Γ is the Lorentz factor of the dissipation region, and c is the velocity of light.Assuming the fraction of the dissipated energy used to accelerate electrons is ε e , one can have ε dis ε e L k = L obs in the fast cooling case, where L obs is the observed luminosity.R dis , Γ, and ò B are degenerate in modeling the magnetic field.Then, the values of ε B = 0.3 (e.g., Lemoine 2013), Γ = 300 (e.g., Yi et al. 2017), ε dis = 0.1, and ε e = 0.1 (e.g., Lemoine 2013) are used in our calculations.
Here, the values of ε dis and ε e do not affect the radiation spectral morphology of the synchrotron emission.
Apart from the decaying random magnetic field generated by the internal shock, there may be a background magnetic field without decay during the shock, e.g., the large-scale magnetic field carried from the central engine of GRBs.We consider internal shocks with a nondecaying background magnetic field ¢ B bg by taking is the ratio of the background magnetic field energy to the initial value of the random magnetic field energy.Therefore, the total magnetic field presented in the downstream region can be described as Electron Prescription and Evolution.It is generally assumed that the accelerated electrons near the shock front obey a power-law distribution, i.e., ( e,0 e,0 0 otherwise, where ( ) with ¢ = ¢ t t 0 , i.e., ( ) g ¢ ¢ t e,0 0 , is the Lorentz factor of electrons injected at the time ¢ t 0 , ( ) e,0 0 is the number of electrons injected in the range [ ( ) 1 is the injection rate of electrons at time ¢ t 0 and is assumed to be the same for different ¢ t 0 in this paper, p is the power-law index, and tot,0 0 e 3 1 2 is the maximum Lorentz factor of the shock-accelerated electrons.Here, m e and q e are the mass and charge of an electron, respectively.The high-energy electrons injected at and near the shock front suffer from radiation cooling by the synchrotron radiation and inverse Compton radiation, i.e., where Y is the Compton parameter.Generally, the radiation cooling of the electrons is strong compared with that due to the expansion of the jet shell in the internal shocks for the prompt emission of GRBs.Then, the adiabatic cooling of electrons is neglected in Equation (5).Since the magnetic field decays with time, the value of Y can be described as where Y 0 denotes the ratio of inverse Compton radiation to synchrotron radiation power at the shock front and Y 0 = 0.5 is set in this paper (Zhao et al. 2014).With Equations (5) and (6), the Lorentz factor of an electron varies with time as where and thus the synchrotron radiation spectrum occurs through the factor 2 .This reveals that the effects of Y 0 and ( ) ¢ ¢ B t tot,0 0 on the synchrotron radiation spectrum are the same if both ¢ B sh,0 and ¢ B bg remain the same in Equation (7).That is to say, the effect of Y 0 on the synchrotron radiation spectrum can be reflected by changing the value of ( ) ¢ ¢ B t tot,0 0 .This behavior has been partially shown in Figure 3 and discussed in Section 4 of Zhao et al. (2014).In this paper, Y 0 = 0.5 is set to simplify our discussion and fittings.
Shock Emission.For electrons injected at time ¢ t 0 , the synchrotron emission power at frequency n¢ and time ¢ t in the comoving frame is given by is the modified Bessel function of order 5/3 and e .The observed spectral flux for an on-axis observer at the observed frequency ν obs is ( ) is the Doppler factor with θ obs = 0 being the viewing angle with respect to the jet axis, Γ = 300 is the Lorentz factor of the downstream region, b = -G 1 1 2 , and d L is the luminosity distance of the burst at redshift z.Since we focus on the radiation spectrum around the peak time of a pulse, the value of ¢ = ¢ = t t 0.3 s end , corresponding to a millisecond pulse in observations, is adopted.

Radiation Spectra of Internal Shocks with a Background Magnetic Field
The radiation spectrum of an internal shock with both a decaying shock-generated random magnetic field and a nondecaying background magnetic field is our focus.Solid lines in Figure 1 show the radiation spectra of an internal shock with different f B , i.e., f B = 0 (black), 10 −4 (magenta), 10 −3 (cyan), 10 −2 (green), and 10 −1 (wine), where L obs = 10 52 erg s −1 , R dis = 10 15 cm, p = 3.5, α B = 1, g ¢ = 3 10 m 3 , and z = 1 are taken. 1Here, g ¢ = 3 10 m 3 and p = 3.5 are adopted to obtain a radiation spectrum (for the case with f B = 0) similar to the Band radiation spectrum with E p = 400 keV, α = − 1, and β = −2.3(gray solid line).One can find that the radiation spectrum from the case with f B = 0 can be similar to the general Band radiation spectrum.This result is consistent with that shown in Zhao et al. (2014).With a nonzero f B , however, a shoulder appears in the low-energy regime.In addition, the shoulder becomes dominant with increasing f B .The shoulder in the low-energy regime is the distinct characteristic of an internal shock with nonzero f B , and The typical isotropic γ-ray luminosity is L obs = 10 51-53 erg s −1 (Zhang 2018) and thus L obs = 10 52 erg s −1 is adopted here.In addition, independent pieces of evidence from different approaches seem to point toward the fact that R dis > 10 13 cm with a typical radius ∼10 15 cm at least for some GRBs (Zhang 2018).Then, R dis = 10 15 cm is adopted.
thus is an important indicator of the background magnetic field in the emission region.It is worth pointing out that a radiation spectrum with a shoulder in its low-energy regime has been found in the prompt emission of some GRBs from optical to MeV bands.
In Figure 1, we also show the power-law radiation spectra with spectral index Ĝ = 1 3 (gray dotted line) or Ĝ = -1 2 (gray dashed-dotted line).According to this diagram, the radiation spectra for a nonzero f B can be decomposed into four segments from the low-energy regime to the high-energy regime, i.e., a / G ^1 3 segment, a is proportional to ν 1/3 .Then, the / G ^1 3 segment in the lowest-energy regime matches the synchrotron radiation spectrum below . Here, { } g ¢ min e,0 is the minimum Lorentz factor of the electrons at the time ¢ t for all of the injected high-energy electrons.(2) The bridge segment makes a bridge between the / G - ^1 2 segment and the G b+ ^1 segment and covers the peak of the νF ν −ν spectrum.From this panel, one can find that the prepeak segments of the νF ν −ν spectra are the same for the case with f B = 0 (black solid line) and for a low f B , e.g., f B = 10 −4 (magenta solid line).It should be noted that the prepeak segment in the f B = 0 case is formed for electrons only in a decaying shock-generated random magnetic field.Then, the bridge segment is formed partially due to the decay of the shock-generated random magnetic field.
(3) Correspondingly, the / G - ^1 2 segment is formed for electrons in a nondecaying background magnetic field.For electrons in the fast cooling environment and a constant magnetic field, the characteristic spectral index of the synchrotron emission is indeed −1/2, which is the same as the spectral index of the / G - ^1 2 segment.(4) The G b+ ^1 segment is formed due to the injection of electrons, and the photon spectral index β is related to the value of p as β −(p + 2)/2.It is worth noting that the photon spectral index β is different for the case with a different f B .This can be found by comparing the wine solid line ( f B = 10 −1 ) with other solid lines in Figure 1.
Other features of the radiation spectrum from our model are as follows.
1. Figure 1 reveals that the spectral index of the bridge segment can vary from Ĝ ~0 to ˆb G = + 1 for the cases of f B varying from ∼0 to 1.In addition, a radiation spectrum with two bumps or a plateau around the peak of the νF ν −ν spectrum may appear if the value of f B is taken appropriately.Since the bridge segment is formed partially due to the decay of the shock-generated random magnetic field, the behavior of this decay may affect the morphology of the bridge segment.In Figure 2(a), we plot the radiation spectrum by adopting different α B , i.e., α B = 0 (black dashed line), α B = 0.5 (red dashed line), α B = 1 (green solid line), α B = 2 (blue dashed line), and α B = 3 (wine dashed line), where the values of other parameters are the same as those adopted for the green solid line.The green solid line is the same as that in Figure 1 and serves as the fiducial case in Figure 2. It should be noted that the case with α B = 0 is equal to the standard straightforward internal shock (standard-IS), of which the shock-generated magnetic field does not decay with time.One can find that the value of α B significantly influences the spectral morphology of the bridge segment.2. In Figure 2(a), one can find that the photon spectral index β of the G b+ ^1 segment is equal to −(p + 2)/2 for the case with α B = 0.However, β is generally larger than −(p + 2)/2 if α B ≠ 0 is adopted.This can be found by comparing the black line (α B = 0) with other lines (α B ≠ 0).This reveals that the decaying behavior of the shock-generated magnetic field also affects the spectral morphology of the G b+ ^1 segment.In Figure 2(b), we plot the radiation spectrum by adopting different p, i.e., p = 2.5 (black dashed line), p = 3 (red dashed line), p = 3.5 (green solid line), and p = 4 (blue dashed line), where the values of other parameters are the same as those adopted for the green solid line.It can be found that the value of β is larger than −(p + 2)/2 in this panel.In addition, β can be larger than −2 if p  2 is adopted, e.g., the case with p = 2.5.The deviation of β from −(p + 2)/2 implies that the power-law index p of the accelerated electrons would be generally higher than −2(β + 1) for the radiation of a shock with a decaying shock-generated magnetic field.3.In Figure 1, the radiation spectrum for the case with f B  0.1 is similar to that for a standard-IS, e.g., the black dashed line in Figure 2.However, it should be noted that the radiation spectrum for the case with both f B  0.1 and α B ≠ 0 is affected by the decaying behavior of the shockgenerated magnetic field.Then, there would be a difference in these two kinds of radiation spectrum.For example, the G b+ ^1 segment deviates from a power-law function for the radiation spectrum for the case with both f B  0.1 and α B ≠ 0. 4. The appearance of a shoulder, i.e., the / G - ^1 2 segment, in the low-energy regime of the radiation spectrum is the main characteristic of an internal shock with a nonzero background magnetic field.Since this segment is mainly due to the cooling of electrons in a constant background magnetic field, the effects on the cooling process, e.g., R dis 2 , g ¢ m , Y 0 , and ¢ t end , would affect this segment.In Figure 2(c), we plot the radiation spectrum by adopting different values of R dis , i.e., R dis = 10 14 (black dashed line), 3 × 10 14 (blue dashed line), 10 15 (green line), 3 × 10 15 (red dashed line), and 10 16 cm (gray dashed line).Here, the values of other parameters are the same as those adopted for the green solid line.By comparing the green solid line and the dashed lines, one can find that if a higher value of R dis or a low strength of magnetic field is adopted, the width of the / G - ^1 2 segment would be narrower.In Figure 2(d), we plot the radiation spectrum by adopting g ¢ = 10 m 4 with a blue dashed line, where the values of other parameters are the same as those adopted for the green solid line.By comparing the green solid line and the blue dashed line, one can find that if a higher value of g ¢ m is adopted, the width of the / G - ^1 2 segment would be broader.This is because a higher value of g ¢ m would create a higher value of { } g g ¢ ¢ min e,0 m at the same time ¢ t .The value of { } g g ¢ ¢ min e,0 m would also be affected by the value of Y 0 and ¢ t end .In Figure 2(d), we plot the radiation spectrum by adopting Y 0 = 2.0 ( ¢ = t 3.0 end ) with a red dashed line (orange dashed line), where other parameters are the same as for the green solid line.This indeed reveals that if a higher value of Y 0 or ¢ t end is adopted, the / G - ^1 2 segment would be broader.It follows that the values of R dis , g ¢ m , Y 0 , and ¢ t end can have an effect on the / G - ^1 2 segment.However, it should be noted again that the effect of Y 0 on the synchrotron radiation spectrum can be reflected by changing the value of ( ) ¢ ¢ B t tot,0 0 based on Equation (7).In Figure 2(d), we also plot the radiation spectrum by adopting τ B = 0.001 (magenta dashed line), which is similar to the wine dashed line (α B = 3) of panel (a).This reveals that the parameters τ B and α B are degenerate in modeling the radiation spectrum of our model.

Spectral Fitting on GRB 211211A
Recently, a peculiarly long-duration gamma-ray burst, GRB 211211A with redshift z = 0.076 (Malesani et al. 2021), was detected by multiple high-energy telescopes, including the Fermi Gamma-Ray Burst Monitor (GBM; Mangan et al. 2021) and the Swift Burst Alert Telescope (D'Ai et al. 2021).GRB 211211A was detected by Fermi-GBM at T 0 = 13:09:59 UT on 2021 December 11 with T 90 ∼ 34.3 s estimated in the energy band of 50-300 keV (Fermi GBM Team 2021).The light curve of prompt emission consists of a hard main emission with a duration ∼12 s and a soft extended emission with a duration ∼50 s.More interestingly, no associated supernova signature was detected, but a kilonova in optical and near-infrared bands is suggested to be associated with GRB 211211A (Rastinejad et al. 2022).The remarkable consistency of these signatures suggests that GRB 211211A originated from the merger of a compact binary (Rastinejad et al. 2022;Gompertz et al. 2023).Gompertz et al. (2023) suggest that the synchrotron emission may be responsible for the pulses in GRB 211211A.In this section, we perform the spectral analysis of the pulses in the hard main emission phase of GRB 211211A based on our model.
We downloaded the time-tagged-event data of GRB 211211A from the public science support center at the Fermi website. 3The brightest NaI and BGO detectors, i.e., n2 and b0, are selected for our analysis.We retrieve GBM spectral data and their corresponding response matrix files (rsp2) from the online HEASARC archive (Gruber et al. 2014;von Kienlin et al. 2014von Kienlin et al. , 2020;;Narayana Bhat et al. 2016).The obtained light curve of the hard main emission of GRB 211211A can be found in Figure 3.In the internal shock scenario, each pulse in the light curve of the prompt emission represents a single excitation by the collision of fast and slow shells to accelerate particles and form emission.Then, we select good sampling pulses from the light curve of the hard main emission.The time slices we selected for our spectral analysis are shown with vertical dashed lines and marked with labels "a"-"p" in Figure 3.For comparison, the Band function, the standard-IS,4 and our model are all used in our spectral analysis, where the Python source package threeML5 (Vianello et al. 2015) is adopted to perform the spectral fittings.There are five free parameters, i.e., α B , p, g ¢ m , f B , and R dis , in our model, 6 where the luminosity L obs = 10 51 erg s −1 is estimated for GRB 211211A.In the standard-IS, there are three free parameters, i.e., p, g ¢ m , and R dis , where α B = 0 and f B = 0 are set.The fitting results estimated based on the maximum likelihood method are reported in Table 1 and the spectral fitting results in the time slice of [6.25, 6.5] s are shown in Figure 4 as an example.
Based on the results shown in Table 1 and Figure 4, we can conclude that our model provides a better fitting for almost all time slices than the Band function.The reasons are as follows.
(1) In Figure 4, the spectral fitting results together with the residuals (σ) are shown at the top and bottom of each panel, respectively.Notably, a good spectral model for the observational data should show a better distribution of the residuals.In the spectral fitting to the observed data with the Band function, which is shown in the left panel of Figure 4, the distribution of the residuals appears as two bumps peaking at around 100 keV and 2000 keV, respectively.In the upper part of the left panel of Figure 4, one can also find that the Band function could not well describe the data, especially those around 100 and 2000 keV.However, the residuals in the spectral fitting to the observed data with our model are well distributed around zero without any feature.This reveals that our model provides a better fitting to the observed data than the Band function.(2) We also estimate the Bayesian information criterion (BIC; Schwarz 1978) for these two models.The BIC is the penalized log-likelihood criterion taking into account the numbers of free parameters, and tends to be one of the most popular criteria for Bayesian model selection.Generally, the model with the lowest BIC is preferred.In the spirit of Burnham & Anderson (2004), the value of ΔBIC can be used as the strength of the evidence to allow quick comparison and ranking of candidate hypotheses or models. 7The values of ΔBIC 1 = BIC Band − BIC Our (first line of each block) and ΔBIC 2 = BIC Standard−IS − BIC Our (second line of each block) from our spectral fittings are also presented in the last column of Table 1.One can find that the values of ΔBIC 1 and ΔBIC 2 obtained from our spectral fittings are in the ranges 7. 87-146.52 and 22.73-37.98,respectively.That is to say, Figure 3.Light curve of the hard main emission of GRB 211211A (black curve) and the time slices, marked with two red vertical dashed lines and labels "a"-"p", for our spectral analysis.both ΔBIC 1 and ΔBIC 2 are significantly larger than 10 for almost all time slices.This implies that for the radiation spectra of our selected time slices, our model is preferred to the Band function and the standard-IS.

Conclusion
Since the jet launched from the central engine of a GRB is generally believed to carry a strong large-scale magnetic field, there may be a large-scale background magnetic field in the emission region.Unlike previous works (e.g., Pe'er & Zhang 2006;Zhao et al. 2014;Wang & Dai 2021), we have taken into account the existence of a background magnetic field under the assumption that the internal shock-generated random magnetic field would decay with time in this paper.With a zero background magnetic field, the synchrotron emission of the internal shock is almost the same as the Band radiation spectrum, which is consistent with that shown in Zhao et al. (2014).However, a shoulder with spectral index −1/2 appears in the low-energy regime of the radiation spectrum with a nonzero background magnetic field.It is found that the shoulder becomes dominant by strengthening the background magnetic field.Correspondingly, a radiation spectrum with two bumps or a plateau around the peak of the νF ν −ν spectrum may appear if the strength of the background magnetic field is given appropriately.Owing to the decay of the shock-generated magnetic field, the radiation spectral morphology in the highenergy regime is not a power-law function even though a power-law distribution of electrons is injected.It is interesting to point out that a radiation spectrum with a shoulder in its lowenergy regime has been found in the prompt emission of some GRBs from optical to MeV bands.
Figure 1 reveals that if the ratio of the background magnetic field energy to the initial value of the shock-generated magnetic field energy f B is around or larger than 0.01, the radiation spectrum of the prompt gamma-ray emission in the low-energy regime (e.g., 10-400 keV) may be different from that of the internal shock without a background magnetic field.Then, we apply our model to fit the radiation spectrum of the pulses in the hard main emission of GRB 211211A, which is suggested to originate from the synchrotron emission (Gompertz et al. 2023).For comparison, we also perform the spectral fitting with the Band function and the synchrotron emission from the standard straightforward internal shock (standard-IS), of which the shock-generated magnetic field does not decay with time.It is found that our model presents a much better fitting to the radiation spectrum than the spectral fitting results with the Band function.Here, the distribution of residuals and the Bayesian information criterion are used as the strength of the evidence for supporting one model over another model.Compared with the standard-IS, our model also presents a better fitting to the radiation based on the Bayesian information criterion.Thus, we conclude that the hard main emission of GRB 211211A may be formed in the internal shocks with both a decaying shock-generated magnetic field and a nondecaying background magnetic field.The ratio of the background magnetic field energy to the initial value of the shock-generated magnetic field energy is around f B ∼ 0.41-0.99based on the spectral fitting results of the pulses in the hard main emission of GRB 211211A.This reveals that the magnetization in the emission region may be moderate in this burst.
b+ ^1 segment, where the G z ^segment represents the spectral index of this segment around ζ, e.g., the spectral index of the / G ^1 3 segment is 1/3.Here, the / G - ^1 2 segment corresponds to the shoulder mentioned above.The origins of these four spectral segments are as follows.(1) For a single electron, the synchrotron radiation spectrum below the characteristic frequency n n = ¢ D c c

Table 1
Spectral Fitting Results of GRB 211211A with the Band Function (First Line of Each Block), the Standard-IS (Second Line), and our Model (Third Line)