Towards a Stringy Description for the $Q\bar Q q\bar q$-Quark System

For the case of two light flavors we propose the stringy description of the system made of one heavy and one light quark-antiquark pair, with the aim of exploring the two lower-lying Born-Oppenheimer potentials as a function of a separation of the heavy quark-antiquark pair. Our analysis reveals three critical separations related to the processes of string reconnection, breaking and junction annihilation. In particular, for the ground state potential only the process of string reconnection matters. We find that a tetraquark state makes the dominant contribution to the potential of the first excited state at small separations, and this is the big difference with the $QQ\bar q\bar q$-quark system where it does so to the ground state potential. Another big difference is the emergence of the full diquark picture $[Qq][\bar Q\bar q]$ rather than the partial picture $QQ[\bar q\bar q]$ for the tetraquark state. On the other hand, the scales of string junction annihilation, below which the systems can be thought of mainly as the compact tetraquarks, are very close for both cases and become almost the same if the phenomenological rule $E_{QQ}=\frac{1}{2} E_{Q\bar Q}$ holds. The same is also true for the screening lengths whose values are in agreement with lattice QCD.