• *Synchrotron Emission*

The synchrotron emission has been extensively studied for first interpreting the nonthermal emission in AGNs and then the GRB afterglow emission [102]. Regarding the explanation of the GRB prompt emission spectra [74,93–95,103], the synchrotron emission model has several advantages:


A source at redshift *z*, expanding with Γ = p 1 − *β* <sup>2</sup> and at an angle *θ* with respect to the observer, emits photons which are seen with a Doppler boost <sup>D</sup> = [Γ(<sup>1</sup> <sup>−</sup> *<sup>β</sup>* cos *<sup>θ</sup>*)]−<sup>1</sup> . In the comoving frame, electrons move in a magnetic field *B* and thus have random Lorentz factor *γ<sup>e</sup>* . Their typical energy is [102]

$$
\varepsilon\_{ob} = \frac{3q\hbar B\gamma\_\varepsilon^2}{2m\_\varepsilon c} \frac{\mathcal{D}}{(1+z)} = 1.75 \times 10^{-19} B\gamma\_\varepsilon^2 \frac{\mathcal{D}}{(1+z)} \text{ erg.} \tag{18}
$$

Typical GRB peak energies *εob* ≈ 200 keV require strong magnetic fields and very energetic electrons, both feasible for Poynting flux-dominated outflows or photon-dominated outflows where strong magnetic fields may be generated via Weibel instabilities [104] 16 .

On the other hand, strong magnetic fields imply the comoving cooling time of the electrons to be *t* 0 *cool* . *t* 0 *<sup>d</sup>* ∼ *R*/(Γ*c*). Thus, the expected synchrotron spectrum below the peak energy would be *F<sup>ν</sup>* ∝ *ν* <sup>−</sup>1/2 (or *N<sup>E</sup>* ∝ *E* <sup>−</sup>3/2) [106,107], which is inconsistent with the average low energy spectral slope h*α*i = −1 (see Figure 2) and, hence, the value *α* = −3/2 is called "synchrotron line of death". To overcome this problem, electrons must cool slowly, leading to a spectrum below the peak given by *F<sup>ν</sup>* ∝ *ν* 1/3 (or *N<sup>E</sup>* ∝ *E* <sup>−</sup>2/3), which is roughly consistent with the observations. However, the condition *t* 0 *cool* & *t* 0 *d* leads to high values of *γ<sup>e</sup>* , whereas *B* would be very low and, in order to explain the observed flux, the electron energy would be several orders of magnitude higher than that stored in the magnetic field [108]. To overcome this, the inverse Compton contribution has to be significant, producing ∼ TeV emission. To avoid a substantial increase of the total energy budget, the emission radius should be *R* & 10<sup>17</sup> cm but cannot explain the rapid variability observed [108].

Suggested modifications (and drawbacks) to the synchrotron scenario can be found in the literature [25,98,109–113].
