Investigations on heavy quarkonium hybrid mesons with exotic quantum numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>C</mml:mi> </mml:mr

We investigate the heavy quarkonium hybrid mesons with exotic quantum numbers J P C = 2 + − via QCD sum rule method. We construct the currents with three Lorentz indices and calculate the correlation functions up to dimension six at the leading order of α s . The states with J P C = 2 + − are extracted by constructing the corresponding projection operators. The obtained results indicate that the masses of 2 + − c ¯ G c and b ¯ G b hybrid states are about 4.49 GeV and 10.48 GeV, respectively. We suggest to search for c ¯ G c hybrid meson with J P C = 2 + − in J / ψf 0,1,2 and χ c 0,1,2 ω final states.