Recently, diffuse and extended sources in TeV gamma-rays as well as in X-rayshave been detected in the direction of the Galactic globular cluster (GC)Terzan 5. Remarkably, this is among the brightest GCs detected in the GeVregime. The nature of both the TeV and the diffuse X-ray signal from Terzan 5is not settled yet. These emissions most likely indicate the presence ofseveral non-thermal radiation processes in addition to these giving rise to theGeV signal. The aim of this work is to search for diffuse X-ray emission from the GeVdetected GCs M 62, NGC 6388, NGC 6541, M 28, M 80 and NGC 6139 to compare theobtained results with the signal detected from Terzan 5. This study will helpto determine whether Terzan 5 stands out amongst other GC or whether a wholepopulation of globular clusters feature similar properties. None of the six GCs show significant diffuse X-ray emission on similar scalesas observed from Terzan 5 above the particle and diffuse galactic X-raybackground components. The derived upper limits allow to assess the validity ofdifferent models that were discussed in the interpretation of themulti-wavelength data of Terzan 5. A scenario based on synchrotron emissionfrom relativistic leptons provided by the millisecond pulsar population can notbe securely rejected if a comparable magnetic field strength as in Terzan 5 isassumed for every GC. However, such a scenario seems to be unlikely for NGC6388 and M 62. An inverse-Compton scenario relying on the presence of aputative GRB remnant with the same properties as the one proposed for Terzan 5can be ruled out for all of the six GCs. Finally, the assumption that each GChosts a source with the same luminosity as in Terzan 5 is ruled out for all GCsbut NGC 6139. (abridged)
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