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Formation of Millisecond Pulsars with Low-Mass Helium White Dwarf Companions in Very Compact Binaries

Binary millisecond pulsars (BMSPs) are thought to have evolved from low-mass X-ray binaries (LMXBs). If the mass transfer in LMXBs is driven by nuclear evolution of the donor star, the final orbital period is predicted to be well correlated with the mass of the white dwarf (WD), which is the degenerate He core of the donor. Here we show that this relation can be extended to very small WD mass ($\sim 0.14-0.17\,M_\odot$) and narrow orbital period (about a few hours), mainly depending on the metallicities of the donor stars. There is also discontinuity in the relation, which is due to the temporary contraction of the donor when the H-burning shell crosses the hydrogen discontinuity. BMSPs with low-mass He WD companions in very compact binaries can be accounted for if the progenitor binary experienced very late Case A mass transfer. The WD companion of PSR J1738+0333 is likely to evolve from a Pop~II star. For PSR J0348+0432, to explain its extreme compact orbit in the Roche lobe-decoupling phase, even lower metallicity ($Z=0.0001$) is required.