Supergiant fast X-ray transients (SFXTs) are a new class of high-mass X-ray binaries (HMXBs) discovered thanks to the monitoring of the Galactic plane performed with the INTEGRAL satellite in the last 5 years. These sources display short outbursts (significantly shorter than typical Be/X-ray binaries) with a peak luminosity of a few 1036 erg s−1. The quiescent level, measured only in a few sources, is around 1032 erg s−1. The X-ray spectral properties are reminiscent of those of accreting pulsars; thus, it is likely that all the members of the new class are indeed HMXBs hosting a neutron star, although only two SFXTs have a measured pulse period, IGR J11215–5952 (~187 s) and IGR J18410–0535 (~4.7 s). Several competing mechanisms have been proposed to explain the shortness of these outbursts, mostly involving the structure of the wind from the supergiant companion. To characterize the properties of these sources on timescales of months (e.g., the quiescent level and the outburst recurrence), we are performing a monitoring campaign with Swift of four SFXTs (IGR J16479–4514, XTE J1739–302, IGR J17544–2619, and AX J1841.0–0536/IGR J18410–0535). We report on the first 4 months of Swift observations, which started on 2007 October 26. We detect low-level X-ray activity in all four SFXTs, which demonstrates that these transient sources accrete matter even outside their outbursts. This fainter X-ray activity is composed of many flares with a large flux variability, on timescales of thousands of seconds. The light-curve variability is also evident on larger timescales of days, weeks, and months, with a dynamic range of more than 1 order of magnitude in all four SFXTs. The X-ray spectra are typically hard, with an average 2-10 keV luminosity during this monitoring of about 1033-1034 erg s−1. We detected pulsations from the pulsar AX J1841.0–0536/IGR J18410–0535, with a period of 4.7008 ± 0.0004 s. This monitoring demonstrates that these transients spend most of the time accreting matter, although at a much lower level (~100-1000 times lower) than during the bright outbursts, and that the true quiescence, characterized by a soft spectrum and a luminosity of a few 1032 erg s−1, observed in the past in only a couple of members of this class, is probably a very rare state.
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