The Pion Gravitational Form Factors and the Trace Anomaly in QCD Factorization

We study the pion gravitational form factor in QCD factorization, focusing on the trace-anomaly component generated by the non-Abelian $TJJ$ vertex. The calculation combines a Sudakov-resummation-improved pion hard kernel with the anomaly form factor suggested by momentum-space conformal field theory and by the perturbative dilaton sum rule. Comparison with lattice QCD data shows a refined projection hierarchy: the isolated anomaly cancels in the form factor $A_{\pi }(Q^2)$, while the full $TJJ$ insertion lowers the leading-order curve at small momentum transfer squared $Q^2$; the anomaly is important in $D_{\pi }(Q^2)$, and it gives the dominant $TJJ$ contribution to the trace form factor.