An evolutionary channel towards the accreting millisecond pulsar SAX J1808.4-3658

Recent timing analysis reveals that the orbital period of the first discovered accreting millisecond pulsar SAX J1808.4-3658 is increasing at a rate $\dot{P}_{\rm orb}=(3.89\pm0.15)\times 10^{-12}~\rm s\,s^{-1}$, which is at least one order of magnitude higher than the value arising from the conservative mass transfer. An ejection of mass loss rate of $10^{-9}~\rm M_{\odot}{\rm yr}^{-1}$ from the donor star at the inner Lagrangian point during the quiescence state could interpret the observed orbital period derivative. However, it is unknown whether this source can offer such a high mass loss rate. In this work, we attempt to investigate an evolutionary channel towards SAX J1808.4-3658. Once the accretion disk becomes thermally and viscously unstable, the spin-down luminosity of the millisecond pulsar and the X-ray luminosity during outbursts are assumed to evaporate the donor star, and the resulting winds carry away the specific orbital angular momentum at the inner Lagrangian point. Our scenario could yield the observed orbital period, the orbital period derivative, the peak X-ray luminosity during outbursts. Low-mass X-ray binaries with a $1.0~\rm M_{\odot}$ donor star, and an orbital period in the range of 0.8 - 1.5 d, may be the progenitor of SAX J1808.4-3658. Our numerical calculations propose that the current donor star mass is $0.044 ~\rm M_{\odot}$, which is in approximately agreement with the minimum mass of the donor star. In addition, our scenario can also account for the formation of black widows or the diamond planet like PSR J1719-1438.