Searching for X-ray counterparts of degree-wide TeV halos around middle-aged pulsars with SRG/eROSITA

Context. Extended gamma-ray TeV emission (TeV halos) has been detected around middle-aged pulsars. A proposed model to explain these TeV halos is that electrons from a degree-wide pulsar wind nebula (PWN) get up-scattered by cosmic microwave background photons through inverse Compton processes. However, no X-ray degree-wide faint diffuse PWNe have been found around these middle-aged pulsars in previous X-ray observations. Aims. We performed a search for degree-wide PWNe around Geminga, PSR B0656+14, B0540+23, J0633+0632, and J0631+1036 using data from the first four consecutive Spectrum Roentgen Gamma/eROSITA all-sky surveys. In order to better understand the mechanisms underlying the formation of TeV halos, we investigated the magnetic field strength within an area of 1 degrees in radius centred on those pulsars. Methods. To achieve our goals, we selected a list of suitable candidate pulsars in the eROSITA-DE part of the sky and applied data reduction techniques to process the eROSITA data. We then performed a spatial analysis of the regions around selected pulsars. Results. We do not detect any degree-wide diffuse emission driven by rotation-powered pulsars around Geminga, PSR B0656+14, B0540+23, J0633+0632, or J0631+1036. Indeed, close inspection of the data shows that the pulsars of interest are all embedded in diffuse emission from supernova remnants such as the Monogem Ring or the Rosetta Nebula, while PSR B0540+23 is located similar to 2.5 degrees away from the Crab pulsar, which is exceptionally bright, causing its X-ray emission to spread over a significant area up to the position of PSR B0540+23 and therefore strongly diminishes our capacity to search for degree-wide bright diffuse X-ray emission. Conclusions. Despite the non-detection of any degree-wide PWN surrounding the analysed pulsars, we set flux upper limits to provide useful information on magnetic field strength and its spatial distribution around those pulsars, providing additional constraints on the proposed theory for the formation of TeV halos around pulsars.