which was unstable in Refs. [55, 56, 54]. It was found that inclusion of the dimension-six gluon condensate 〈 g $^{3}$G$^{3}$〉 stabilizes the sum rule in this channel, permitting reliable mass pre- dictions to be made. The publications in this chapter update the J P C = 1 ++ and 0 − + heavy quarkonium hybrid sum rules to include the effects of the dimension-six gluon condensate. The axial vector (1 $^{++}$) and pseudoscalar (0 − $^{+}$) heavy quarkonium hybrids can be studied within QSR using the following current and correlation function [55, 56, 54]: where q is the external momentum, Q denotes a heavy (charm or bottom) quark field and G a µν denotes the gluon field strength tensor. Because the correlation function is Lorentz invariant, it can be decomposed into where Π$_{V}$ ( q $^{2}$) and Π$_{S}$ ( q $^{2}$) couple to axial vector and pseudoscalar heavy quarkonium hybrids, respectively [54]. These functions can be isolated by contracting Eq. (3.3) with appropriate combinations of the metric and the external momentum. Note that the contractions must be performed in d -dimensions when dimensional regularization is used. The axial vector channel sum rule analysis resulted in mass predictions of 5 . 13 ± 0 . 25 GeV and 11 . 32 ± 0 . 32 GeV for the charmonium and bottomonium hybrids, respectively. Interest- ingly, the dimension-six gluon condensate had little effect on the sum rules in this channel. Although the axial vector channel sum rule was stable in the original analysis [55, 56, 54], it was important to examine the effects of the dimension-six condensate on this channel. Furthermore, because the quantum numbers of the X (3872) are now firmly established to be 1 ++ [2], clear mass predictions for all axial vector exotic hadrons are needed. As dis- 3.2 Results Π$_{µν}$ ( q ) = i$^{∫}$ d $^{4}$x e iq · $^{x}$〈 Ω | T [ J$_{µ}$ ( x ) J$_{ν}$ (0) ] | Ω 〉 , (3.1) J$_{µ}$ = g 2 ¯ Qλ $^{a}$γν ˜ G a $_{µν}$Q , ˜ G a µν = 1 2 ϵ$_{µναβ}$G αβ a , (3.2) Π$_{µν}$ ( q ) = $^{[}$q$_{µ}$ q$_{ν}$ q 2 − g$_{µν}$$^{]}$ Π$_{V}$ $^{(}$q 2 $^{)}$+ q$_{µ}$q$_{ν}$ q 2 Π$_{S}$ $^{(}$q 2 ) , (3.3) 79