Understanding the evolution of black hole spin in X-ray binary systems: Case study of XTE~J1550-564

We present a comprehensive study of the X-ray binary system XTE~J1550-564, with the primary objective of analyzing the evolution of the black hole's spin parameter. To achieve this objective, we embarked on the necessary step of identifying a plausible progenitor for the system. Using a set of models covering various parameter combinations, we were able to replicate the system's observed characteristics within acceptable error margins, including fundamental parameters such as component masses, orbital period, donor luminosity, and effective temperature. The model results indicate the possibility of diverse evolutionary pathways for the system, highlighting the significant role played by the initial mass of the donor star and the efficiency of mass transfer episodes. While some models are well-aligned with estimates of the mass transfer rate, they all fall short of explaining the black hole's observed moderate spin ($a^* = 0.49$). We also explored alternative magnetic braking prescriptions, finding that only an extreme and fully conservative scenario, based on the convection and rotation boosted prescription, can reproduce the observed spin and only in a marginal way. Our study attempts to shed light on the complex dynamics of black hole X-ray binaries and the challenges of explaining their observed properties with theoretical models.