Perturbative QCD constraints on the gravitational form factors

We discuss the gravitational form factors (GFFs) using the QCD constraints on the hadron matrix elements of the energy-momentum tensor T^μν. With those constraints taking into account the perturbation contributions, the asymptotic behaviors of the GFFs at large momentum transfers and the forward values of the GFFs at zero momentum transfer are discussed. In particular, the latter results are derived for the twist-four gravitational form factor for the quark as well as for the gluon, C_q,g, at NNLO QCD from the trace anomaly constraints. For this purpose, the QCD trace anomaly is attributed to the anomalies for each of the gauge-invariant quark part and gluon part of T^μν, and those anomalies are evaluated at the three loop order. For the nonperturbative matrix elements arising in the corresponding formulas, we use the present information from experiment and lattice QCD. We show quantitative results for the nucleon as well as for the pion, leading to a model-independent determination of the forward value of C_q,g. We also mention that the same framework based on the trace anomaly constraints also indicates quite different pattern in the mass structures between the nucleon and the pion.