*3.5. Maximum Synchrotron Photon Energy*

One of the expectations from Fermi-LAT observations of GRB afterglows was the identification of a spectral cutoff in the afterglow synchrotron spectrum marking the maximum energy of synchrotron photons [122,123]. Such a cutoff has not been firmly identified. Its location is directly connected with the shock micro-physics conditions and the maximum energy at which electrons can be accelerated. This maximum energy is typically estimated equating the timescale for synchrotron cooling and the acceleration timescale, where acceleration is assumed to proceed at the Bohm level, considered as the fastest rate. This estimate returns, hence the maximum energy of the accelerated particles. Assuming that the accelerated particles are electrons:

$$t\_L' = \frac{r\_L}{c} = \frac{\gamma m\_\epsilon c}{eB'} \tag{53}$$

where *r<sup>L</sup>* is the Larmor radius. For each crossing, the electrons gain energy by a factor ∼ 2. On the other hand, the energy losses for synchrotron radiation on this timescale are:

$$\delta E' = t\_L' P' = \frac{\gamma m\_\varepsilon c}{eB'} \frac{\sigma\_T c \gamma^2 B'^2}{6\pi} = \frac{1}{6\pi} \frac{\sigma\_T m\_\varepsilon c^2 B' \gamma^3}{e} \tag{54}$$

The particle stops to gain energy when:

$$
\delta E' = \gamma m\_\varepsilon c^2 \tag{55}
$$

Therefore, the maximum Lorentz factor for electrons can be derived:

$$
\gamma\_{\text{max}} = \sqrt{\frac{3\pi e}{\sigma\_T B'}}\tag{56}
$$

The corresponding maximum synchrotron photon energy is:

$$h\nu\_{\text{max}}^{\prime} = \frac{eB^{\prime}\gamma\_{\text{max}}^2h}{2\pi m\_{\text{eff}}c} \tag{57}$$

which for electrons is ∼50 MeV in their rest frame.

Similar considerations can also be conducted for protons. Following the same arguments presented below, one obtains:

$$\gamma\_{\rm cool,p} = \frac{6\pi m\_p^3 c}{\sigma\_\Gamma m\_\varepsilon^2 B'^2 t'} \tag{58}$$

for the cooling Lorentz factor and:

$$\gamma\_{\text{max},p} = \sqrt{\frac{3\pi em\_p^2}{\sigma\_T m\_e^2 B'}}\tag{59}$$

for the maximum Lorentz factor, which sets a maximum photon energy of ∼100 GeV. Synchrotron emission is less efficient for protons so they are less affected by cooling and they can reach higher maximum Lorentz factors than the electrons.

Within this framework, it is expected that observations in the GeV band can be exploited to identify the presence of a cut-off in the HE tail domain. At the current stage, Fermi-LAT observations indicate that the afterglow component of the HE energy emission is usually modeled with a single power-law component with index ∼−2 and with no evidence of spectral evolution in time [124] and HE cut-offs.

The absence of the cut-off in the observational data may be explained in several ways. The most likely interpretations are the limited sensitivity of the LAT instrument in the GeV range and the possible contamination due to the rising of the SSC spectral component. As a result, the synchrotron cut-off is hidden behind the VHE spectral component, which can be detected in the GeV-TeV domain. This implies that TeV observations are fundamental in order to disentangle between the two spectral components and infer the cutoff of the synchrotron spectrum.

Another possible interpretation is that the lack of a cutoff in the observational data is genuine. In this case, the synchrotron emission can exceed the limit assumed for the maximum photon energy and extend in the GeV-TeV domain. This interpretation can be tested with VHE observations. The extension of the HE power-law derived by LAT up to the TeV domain should be consistent with VHE data, and no spectral hardening in the GeV-TeV band should be observed. Such a scenario is in contrast with the standard particle acceleration model presented in Section 2. A deep revision of our understanding of acceleration mechanisms is required in order to make TeV emission from synchrotron radiation possible. In particular, a mechanism, able to accelerate electrons up to PeV energies, is needed.

The calculation performed so far assumes the presence of a uniform magnetic field *B* 0 throughout the shock-heated plasma. If this assumption is rejected, it is possible to consider a non-uniform magnetic field, stronger close to the shock front and decaying downstream. Following the calculation of [125]. the magnetic field *B* 0 can be expressed in terms of the distance from the shock front *x* as:

$$B'(\mathbf{x}) = B\_s' \left(\frac{\mathbf{x}}{L\_p}\right)^{-\eta} + B\_w' \tag{60}$$

where *B* 0 *<sup>s</sup>* and *B* 0 *<sup>w</sup>* are, respectively, the strongest and the weakest magnetic field strengths, *η* is the power-law decaying index, and *L<sup>p</sup>* is the field decay length scale, which is estimated as [126]:

$$L\_p = \left[\frac{m\_p \Gamma\_s c^2}{4\pi n e^2}\right]^{1/2} = 2.2 \times 10^7 \frac{\Gamma\_s}{n} \quad \text{cm} \tag{61}$$

where Γ*<sup>s</sup>* is the shock front Lorentz factor and *n* is the number density of the accelerated particles in the shocked fluid comoving frame. As a consequence, the Larmor radius *r<sup>L</sup>* increases with the distance from the shock front, since *B* 0 (*x*) becomes weaker and an electron travelling downstream will be likely sent back upstream when *r<sup>L</sup>* ≤ *x*. When considering the case *B* 0 *<sup>s</sup> B* 0 *<sup>w</sup>* and *x L<sup>p</sup>* the particles will lose most of their energy in the region of low magnetic field. Therefore, from the condition that losses in the low magnetic field region should be greater than losses in the high magnetic field region, after some algebra the following condition is obtained:

$$\left(\frac{B\_s'}{B\_w'}\right)^2 \lesssim \frac{r\_l}{L\_p} \tag{62}$$

valid for *η* > 1/2 and *x*0/*L<sup>p</sup>* 1 where *x*<sup>0</sup> is the width of the high magnetic field region. Considering that *x*0/*L<sup>p</sup>* ≡ (*B* 0 *<sup>s</sup>*/*B* 0 *w*) 1/*η* , Equation (62) states that the Larmor radius in the high magnetic field region is larger than the actual width of the region and electrons will be barely deflected in such portion of the shocked plasma. As a result, it is possible to calculate the maximum Lorentz factor for electrons that lose most of their energy in the weak magnetic field region following the same conditions presented for the uniform magnetic field case:

$$
\gamma\_{\text{max}} = \sqrt{\frac{3\pi e}{\sigma\_T B\_w'}}\tag{63}
$$

As a result, the maximum synchrotron photon energy is given by:

$$h\nu\_{\text{max}}' = \frac{e\gamma\_{\text{max}}^2 h}{2\pi m\_e c} \left(\frac{B\_s'}{B\_w'}\right) \tag{64}$$

which is greater than the one calculated in Equation (57) by a factor *B* 0 *<sup>s</sup>*/*B* 0 *<sup>w</sup>*. Numerical calculations [41] show that this ratio can be larger than <sup>∼</sup>10<sup>2</sup> . As a result, photons of energies & 100 GeV can be produced via synchrotron process when assuming a nonuniform magnetic field, which decays downstream of the shock front and with particles losing most of their energies in the weakest field region.

In both interpretations presented here, TeV observations are fundamental in order to investigate with unprecedented details the possible presence or absence of the synchrotron cutoff spectrum. This have also a direct impact on the study of the possible radiation mechanisms responsible for the VHE component in GRBs.
