*2.3. What Radiation Mechanism Produces the Band-like GRB Spectrum?*

Few radiation mechanisms have been proposed to explain the Band-like spectrum of prompt emission, the most popular being synchrotron and inverse-Compton. Below, we present a concise summary of the different proposed mechanisms and show the expected polarization in Figure 1.

#### 2.3.1. Optically-Thin Synchrotron Emission

Relativistic electrons gyrating around magnetic field lines cool by emitting synchrotron photons. When the energy distribution of these electrons is described by a power law, e.g., that obtained at collisionless internal shocks due to Fermi acceleration, the emerging synchrotron spectrum is described by multiple power-law segments that join at characteristic break energies [87,88]. These correspond to the synchrotron frequency, *E<sup>m</sup>* = Γ(1 + *z*) <sup>−</sup>1*hν* 0 *<sup>m</sup>* = Γ(1 + *z*) <sup>−</sup>1*γ* 2 *<sup>m</sup>*(*heB* ¯ 0/*mec*), of minimal-energy electrons with LF *γ<sup>m</sup>* and the cooling frequency, *E<sup>c</sup>* = Γ(1 + *z*) <sup>−</sup>1*hν* 0 *<sup>c</sup>* = 36*π* 2 (1 + *z*) −1 (*hem*¯ *<sup>e</sup>c* <sup>3</sup>/*σ* 2 *T* )(Γ 3*β* <sup>2</sup>/*B* <sup>0</sup>3*R* 2 ), of electrons that are cooling at the dynamical time, *t* 0 cool = *t* 0 dyn = *R*/Γ*βc*. Here, *B* 0 is the comoving magnetic field, and *σ<sup>T</sup>* is the Thomson cross-section. The high radiative efficiency of prompt emission demands that the electrons be in the fast-cooling regime for which *E<sup>c</sup>* < *E<sup>m</sup>* and the *νF<sup>ν</sup>* spectrum peaks at *E*pk = *Em*. In this case, the spectrum below the peak energy has a photon index *α*ph = −2/3 for *E* < *E<sup>c</sup>* and *α*ph = −3/2 for *E<sup>c</sup>* < *E* < *Em*. Above the peak energy, the photon index is *α*ph = −(*p* + 2)/2 where *p* is the power-law index of the electron distribution.

**Figure 1.** Approximate degree of polarization for different radiation mechanisms and jet structures [24]. If the emission is synchrotron then polarization for different B-field configurations is given (assuming Πmax = 70%). For each jet structure, a distinction is made between two cases: (i) when the observer's viewing angle (*θ*obs) is much smaller than the half-opening angle (*θ<sup>j</sup>* ) of a top-hat jet or, in the case of a structured jet, if it is much smaller than the core angle (*θc*) and (ii) when *θ*obs is close to *θ<sup>j</sup>* , i.e., the edge of the jet. For a structured jet, *θ*obs can exceed *θ<sup>c</sup>* by an order unity factor before the fluence starts to drop significantly. When *θ*obs ≈ *θ<sup>j</sup>* , the minimum value of polarization can be zero in all cases, except for *B*tor, for a pulse with a given *ξ<sup>j</sup>* = (Γ*θ<sup>j</sup>* ) 2 , where *ξ* 1/2 *j* is the ratio of the angular sizes of the jet and of the beaming cone. Different pulses may have slightly different *ξ<sup>j</sup>* (typically with a similar *θ<sup>j</sup>* but different Γ), which on average would yield a finite polarization. The quoted lower range reflects this mean value (see [24] for more details). For the *B*tor case, Π = 0% when *θ*obs = 0 due to symmetry and |Π| > 0% otherwise, while Π ≈ 50% at 1/Γ < *θ*obs < *θ<sup>j</sup>* .

While synchrotron emission is still regarded as the default emission mechanism, the basic "vanilla" model has been argued to be not as robust as previously thought. First, its predictions have been challenged by a small fraction of GRBs that showed harder low-energy (*E* < *E*pk) spectral slopes with *α*Band > −2/3 [89–92], often identified as the synchrotron *line of death*. Some possible alternatives that have been suggested to resolve this discrepancy include anisotropic electron pitch angle distribution and synchrotron self-absorption [93], jitter radiation [94], and photospheric emission [95]. The line-ofdeath violation is generally derived by fitting the empirical Band-function to the observed spectrum. When synthetic synchrotron spectra (after having convolved with the energy response of a detector) are fit with the Band-function, an even softer h*α*Bandi = −0.8 is found due to the detector's limited energy range (e.g., *Fermi*/GBM [96]), which does not quite probe the asymptotic value of *α*ph. Since a significant fraction of GRBs show low-energy spectral indices that are harder than this value, it might indicate that another spectral component is possibly contributing at low energies and offsetting the spectral slope. Second, the spectral peak energy in the cosmological rest-frame of the source is given by *Em*(1 + *z*), which depends on a combination of Γ, *γm*, and *B* 0 to yield the measured peak energy in the range 200 keV . [*E*pk,z = *E*pk(1 + *z*)] . 1 MeV [97] with a possible peak around *E*pk,z ∼ *mec* 2 [98]. Given that all of these quantities can vary substantially between different bursts, the synchrotron model does not offer any characteristic energy scale at which most of the energy is radiated in the event that the *E*pk,z distribution indeed narrowly peaks around ∼ *mec* 2 [99]. Third, the synchrotron model predicts wider spectral peaks than that obtained by fitting the Band-function to observations [100]. This issue has now been demonstrated for a large sample of GRBs where the spectral widths obtained with the simplest synchrotron model yielding the narrowest spectral peak, e.g., a slow-cooling Maxwellian distribution of electrons, is inconsistent with most of the GRBs [101,102].

Moreover, it is rather easy to get a wider spectral peak by having, e.g., fast-cooling particles, variable magnetic fields, etc., but it is much harder to obtain narrower peaks.

Several works that find the synchrotron model to be inconsistent with observations invariably use empirical models, e.g., the Band-function, a smoothly-broken power law, to determine low-energy spectral slopes and peak widths. This may become a problem in instances where such models are unable to capture the intrinsic complexity of the underlying data. Therefore, an arguably better approach is to directly fit physical models to the raw data to derive spectral parameters and remove any bias [103–106]. Such an approach has led to alleviating some of the issues encountered by the optically thin synchrotron model, where it was shown that direct spectral fits (in count space rather than energy space) with synchrotron emission from cooling power-law electrons can explain the low-energy spectral slopes as well as the spectral width of the peak [107].
