Prospects for the detection of synchrotron halos around middle-age pulsars

The origin of cosmic-ray positrons detected with an energy above 10 GeV is one of the most intriguing mysteries in Astroparticle Physics. Different interpretations have been invoked to solve this puzzle such as pulsar wind nebulae (PWNe), supernovae remnants or interactions of dark matter particles. The HAWC and Milagro experiments have measured an extended emission of photons above 10 TeV in the direction of Geminga and Monogem PWNe which can be interpreted by photons emitted through inverse Compton scattering by positrons and electrons accelerated by these sources. These HAWC and Milagro detections are extremely important to estimate the PWN contribution to the positron excess. Positrons and electrons emitted by PWNe and injected in the interstellar medium can produce photons with an energy between radio and X-ray through synchrotron radiation with the Galactic magnetic field. In this white paper we show that Astrophysics Probe mission concepts, such as AMEGO (the All-sky Medium Energy Gamma-ray Observatory) and AdEPT (The Advanced Energetic Pair Telescope), are suitable to detect the synchrotron halos from Monogem and Geminga PWNe and we will report the list of the most promising PWNe that AMEGO could be able to detect.