*3.6. Prompt Emission Efficiency*

The overall efficiency *η<sup>γ</sup>* of the prompt emission mechanism is the result of three processes: the efficiency *ηdiss* of the (still unidentified) mechanism responsible for the dissipation of the jet energy, the efficiency *e<sup>e</sup>* of the acceleration mechanism in converting the dissipated energy into random energy of the electrons, and the radiative efficiency *erad* of the electrons: *η<sup>γ</sup>* = *ηdisseeerad*. Provided that it is reasonable to assume a fast cooling regime for the prompt emission (*erad* = 1), the overall prompt efficiency is limited by the capability of the dissipation mechanism in extracting the kinetic or magnetic energy of the jet and the capability of the particle acceleration process to convey a fraction of this energy into the non-thermal electron population. The value of the efficiency then provides a fundamental clue to placing constraints on the origin of energy dissipation in GRBs, which is still quite uncertain, discriminating between matter and magnetic dominated jets.

From the definition of *η<sup>γ</sup>* = *Eγ*/*E*<sup>0</sup> (where *E*<sup>0</sup> = *E<sup>γ</sup>* + *E<sup>k</sup>* is the initial explosion energy), we can write the relation *E<sup>k</sup>* = (1 − *ηγ*)/*ηγEγ*. The parameter *η<sup>γ</sup>* can then be estimated from the comparison of the energy *E<sup>γ</sup>* emitted in the prompt phase and the energy *E<sup>k</sup>* left in the jet after the end of the prompt emission (i.e., at the beginning of the afterglow phase). While the former is directly estimated from observations, the latter can be inferred only indirectly, from the modeling of afterglow radiation.

One of the most adopted methods to infer *E<sup>k</sup>* for large samples of GRBs is to rely on the X-ray luminosity and use it as a proxy for the energy content of the blast-wave [127–131]. This method is solid as long as the X-ray band lies above *max*(*νm*, *nuc*) and is not affected by inverse Compton cooling. If these two conditions are verified, then the electrons emitting X-ray photons are in a fast cooling regime and their cooling is dominated by synchrotron losses. The luminosity produced is then proportional to the energy content of the accelerated electrons *Eke<sup>e</sup>* . Assuming a value (typically 0.1) for *e<sup>e</sup>* , then *E<sup>k</sup>* can be estimated. Investigations based on the X-ray emission have inferred very large values of *ηγ*, between 0.5 and 0.9 [49,129,132,133].

The very same approach can also be applied to 100 MeV-GeV photons detected by the LAT, under the assumption that these are synchrotron photons. A strong correlation between the GeV luminosity and *Eγ*,*iso* has been indeed found, supporting the possibility that GeV photons lie in the high-energy part of the synchrotron spectrum, where the afterglow luminosity is proportional to *Eke<sup>e</sup>* and can be used, similarly to the X-ray luminosity, to estimate *E<sup>k</sup>* [134]. A study by [70] revealed that the energetics *E<sup>k</sup>* inferred independently from X-ray and GeV luminosities on a sample of 10 GRBs are inconsistent with each other. The authors show that the inconsistency is solved if *ν<sup>c</sup>* > *ν<sup>X</sup>* (where *ν<sup>X</sup>* is the X-ray frequency), or if Compton losses are important in the X-ray band. Full modeling of the GeV, X-ray and optical data support this scenario. In both cases, *e<sup>B</sup>* is required to be much smaller than usually assumed, with values in the range 10−7–10−<sup>3</sup> . This analysis shows that the GeV band is a much better proxy for *E<sup>k</sup>* , since it is above *ν<sup>c</sup>* and is not affected by inverse Compton cooling, due to Klein–Nishina suppression. Adopting GeV luminosities as a proxy for *E<sup>k</sup>* , the estimated values of *E<sup>k</sup>* increase, also affecting the estimates of *ηγ*, which are around 5–10% [24].

A correct estimate of *η<sup>γ</sup>* is extremely important, since its value is related to the mechanism dissipating energy in the jet. Since internal shocks can hardly reach values of *η<sup>γ</sup>* larger than 10%, values around 50–90% have been used to argue that internal shocks are not a viable mechanism to explain prompt emission in GRBs, and more efficient mechanisms should be considered (e.g., magnetic reconnection). If the efficiency is, however, smaller than initially estimated, internal shocks may still be a viable solution. Moreover, different estimates of *η<sup>γ</sup>* lead to different estimates on the total initial jet energy *E*<sup>0</sup> = *Eγ*,*iso* + *E<sup>k</sup>* , with repercussions on the energy budget of GRBs and finally on their progenitors and mechanisms for jet launching. Small values of *e<sup>B</sup>* may then relax the problem with very large prompt efficiency, which is definitely unreasonable for internal shocks, but also difficult to attain for magnetic reconnection models (for a discussion, see e.g., [19,20,135]).

A scenario where the magnetic field strength is relatively low in the emitting region implies a stronger SSC emission. Recent TeV detection of GRBs support this scenario, and provide additional observations to constrain the magnetic field. Moreover, as shown by the first detections by IACTs (Section 4), the energy in the TeV component is comparable to the energy in X-rays, providing better estimates for the energy budget in the afterglow phase. Future detections from a larger sample of GRBs can help in assessing the energy budget of the jet more precisely during the afterglow emission, add important information to constrain the efficiency of the prompt emission and favor or exclude some dissipation scenarios.
