Abstract The magnetic braking (MB) mechanism plays a vital role throughout the evolution of low-mass X-ray binaries (LMXBs). Considering the standard MB prescription, the initial orbital periods of LMXBs that can evolve into binary millisecond pulsars (MSPs) with He white dwarfs (WDs) and short orbital periods (2–9 hr) are within an extremely narrow interval that was named the fine-tuning problem. Employing the detailed binary evolution model, we investigate the evolution of LMXBs in both the standard and convection and rotation-boosted (CARB) MB laws. In the standard MB case, it is difficult for donor stars to form a He core and exhaust the H envelope through mass transfer at short orbital periods, making them semidetached systems. The CARB MB mechanism can drive LMXBs to evolve toward compact detached MSP–WD systems in wide initial orbital periods over which binary MSPs with long orbital periods will be produced. We obtain the initial parameter space of binary MSPs with He WDs in the initial orbital period and donor-star mass plane, which can be applied to future statistics study by population synthesis simulations. We also discuss a new relation between orbital period and WD mass, the formation of persistent ultracompact X-ray binaries with relatively long orbital periods, and the detectability of compact MSP–WD systems as low-frequency gravitational-wave sources.
Yang et al. (2025) studied this question.