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Context. Extended gamma-ray TeV emission (TeV halos) around middle-agedpulsars has been detected. A proposed model to explain these TeV halos is thatelectrons from a degree-wide Pulsar Wind Nebula (PWN) get up-scattered bycosmic microwave background photons through inverse Compton processes. However,no X-ray degree-wide faint diffuse PWNe have been found around thesemiddle-aged pulsars in previous X-ray observations.  Aims. We have performed a search for degree wide PWNe around Geminga, PSRB0656+14, B0540+23, J0633+0632, and J0631+1036, using data from the first fourconsecutive Spectrum Roentgen Gamma/eROSITA all-sky surveys. In order to betterunderstand the mechanisms underlying the formation of TeV halos, weinvestigated the magnetic field strength in the degree wide neighbourhood ofthose pulsars.  Results. We did not detect degree-wide diffuse emission around Geminga, PSRB0656+14, B0540+23, J0633+0632, and J0631+1036, which can be attributed tobeing powered by the rotation-powered pulsars. Indeed, a close inspection ofthe data shows that the pulsars of interest are all embedded in diffuseemission from supernova remnants like the Monogem Ring or the Rosetta Nebula,while PSR B0540+23 is located ~2.5 degrees away from the bright Crab pulsar,which shines out the eROSITA point-spread function up to the position of PSRB0540+23 and thus reduced the sensitivity to search for degree wide brightdiffuse X-ray emission strongly.  Conclusions. Despite the non-detection of any degree-wide PWN surrounding theanalysed pulsars, we set flux upper limits to provide useful information onmagnetic field strength and its spatial distribution around those pulsars,providing additional constraints to the proposed theory for the formation ofTeV halos around pulsars.

Searching for X-Ray Counterparts of Degree Wide TeV Halos Around Middle-Aged Pulsars with SRG/eROSITA