Hybrid mesons in the context of the relativistic equation

The flux tube model of hybrid mesons is studied in the context of the relativistic equation in the adiabatic approximation. The moment of inertia of the rigid-rod flux tube is considered in the kinetic part of the interaction. The nonrelativistic and relativistic one scalar bead flux tube is integrated numerically and compared with the adiabatic flux tube small oscillation approximation. The relativistic scalar bead picture suggests that the lowest gluonic excitation of a massive gluon is color octet q q? pair while the conventional mesons are color singlet as usual. We estimate the lowest hybrid charmonium states as 1₁ P = 4.17 ± 0.03 and 1₁D = 4.39±0.05 GeV with color octet qq pair and as 1₁P = 3.78± 0.05 and 1₁D = 4.04±0.09 GeV with color singlet q q? pair. These results are also confirmed by the adiabatic flux tube small oscillation approximation when the hybrid energy gap is adjusted phenomenologically to M _Hc = M _ground + 1GeV.