Body-centred-cubic (BCC) transition metals (TMs) tend to be brittle at low temperatures, posing significant challenges in their processing and major concerns for damage tolerance in critical load-carrying applications. The brittleness is largely dictated by the screw dislocation core structure; the nature and control of which has remained a puzzle for nearly a century. Here, we introduce a universal model and a physics-based material index χ that guides the manipulation of dislocation core structure in all pure BCC metals and alloys. We show that the core structure, commonly classified as degenerate (D) or non-degenerate (ND), is governed by the energy difference between BCC and face-centred cubic (FCC) structures and χ robustly captures this key quantity. For BCC TMs alloys, the core structure transition from ND to D occurs when χ drops below a threshold, as seen in atomistic simulations based on nearly all extant interatomic potentials and density functional theory (DFT) calculations of W-Re/Ta alloys. In binary W-TMs alloys, DFT calculations show that χ is related to the valence electron concentration at low to moderate solute concentrations, and can be controlled via alloying. χ can be quantitatively and efficiently predicted via rapid, low-cost DFT calculations for any BCC metal alloys, providing a robust, easily applied tool for the design of ductile and tough BCC alloys.