Orbital evolution of an accreting millisecond pulsar: witnessing the banquet of a hidden black widow?

We have performed a timing analysis of all the four X-ray outbursts from the accreting millisecond pulsar SAX J1808.4-3658 observed so far by the Proportional Counter Array on board the Rossi X-ray Timing Explorer. For each of the outbursts, we derived the best-fitting value of the time of ascending node passage. We find that these times follow a parabolic trend, which gives an orbital-period derivative P(orb) = (3.40 +/- 0.18) x 10(-12)ss(-1), and a refined estimate of the orbital period, P(orb) = 7249.156 499 +/- 1.8 x 10(-5) s (reference epoch T(0) = 509 14.8099 MJD). This derivative is positive, suggesting a degenerate or fully convective companion star, but is more than one order of magnitude higher than what is expected from secular evolution driven by angular momentum losses caused by gravitational radiation under the hypothesis of conservative mass transfer. Using simple considerations on the angular momentum of the system, we propose an explanation of this puzzling result assuming that during X-ray quiescence the source is ejecting matter (and angular momentum) from the inner Lagrangian point. We have also verified that this behaviour is in agreement with a possible secular evolution of the system under the hypothesis of highly non-conservative mass transfer. In this case, we find stringent constraints on the masses of the two components of the binary system and its inclination. The proposed orbital evolution indicates that in this kind of sources the neutron star is capable to efficiently ablate the companion star, suggesting that this kind of objects are part of the population of the so-called black widow pulsars, still visible in X-rays during transient mass-accretion episodes.