Internal color contributions to flux tube entanglement entropy

In recent work arXiv:2410.00112, we introduced and computed entanglement entropy of the color flux tube (FTE$^2$) between a heavy quark-antiquark pair in (2+1)D Yang-Mills theory. Our numerical results suggest that FTE$^2$ can be partitioned into a component corresponding to transverse vibrations of the flux tube and an internal color entropy. Further, motivated by analytical (1+1)D calculations, and SU(2) (2+1)D Yang-Mills numerical results, we argued that the internal entropy takes the form $\langle F\rangle\log(N_c)$, with $\langle F\rangle$ the number of times, on average, that the flux tube crossed a boundary between region $V$ and its complement. We extend here our FTE$^2$ study to consider different geometries of region $V$, varying the number of boundary crossings, and number of colors. Our preliminary results support the conjectured form of the internal entropy, albeit with noteworthy subtleties relating to the partial/full intersections of the flux tube with the region $V$.