A Dilaton Sum Rule for the Conformal Anomaly Form Factor in QCD at Order $\alpha_s$

We present an off-shell dispersive analysis of the graviton-gluon-gluon ($TJJ$) vertex, extending previous investigations carried out in both QED and QCD. Within the framework of a non-Abelian gauge theory, we extract the conformal anomaly form factor from the trace component of the correlator and demonstrate that it satisfies a one-loop sum rule, valid under the most general kinematic conditions. Analogously to the chiral and chiral-gravitational cases, a spectral flow emerges in which the exchanged intermediate state becomes localized at zero invariant mass along the graviton line as the quark mass approaches zero. The total integral of the spectral density precisely reproduces the anomaly. We examine how the behaviour of these spectral densities evolves as the system approaches the conformal limit with on-shell gluons. The perturbative analysis reveals that such sum rules are fundamental dynamical features of anomaly-induced interactions. In particular, the appearance or absence of associated dilaton poles is closely tied to whether the sum rule is saturated by a pole contribution or by a dispersive continuum. In the conformal, on-shell limit, the particle-pole interaction yields a nonlocal S-matrix element entirely supported on the light-cone.