  Massive runaway stars produce bow shocks through the interaction of their
winds with the interstellar medium, with the prospect for particle acceleration
by the shocks. These objects are consequently candidates for non-thermal
emission. Our aim is to investigate the X-ray emission from these sources. We
observed with XMM-Newton a sample of 5 bow shock runaways, which constitutes a
significant improvement of the sample of bow shock runaways studied in X-rays
so far. A careful analysis of the data did not reveal any X-ray emission
related to the bow shocks. However, X-ray emission from the stars is detected,
in agreement with the expected thermal emission from stellar winds. On the
basis of background measurements we derive conservative upper limits between
0.3 and 10 keV on the bow shocks emission. Using a simple radiation model,
these limits together with radio upper limits allow us to constrain some of the
main physical quantities involved in the non-thermal emission processes, such
as the magnetic field strength and the amount of incident infrared photons. The
reasons likely responsible for the non-detection of non-thermal radiation are
discussed. Finally, using energy budget arguments, we investigate the
detectability of inverse Compton X-rays in a more extended sample of catalogued
runaway star bow shocks. From our analysis we conclude that a clear
identification of non-thermal X-rays from massive runaway bow shocks requires
one order of magnitude (or higher) sensitivity improvement with respect to
present observatories.
