Gravitational $D$-Form Factor: The $\sigma$-Meson as a Dilaton confronted with Lattice Data

We investigate the nucleon and pion gravitational $D$-form factors, by fitting a $\sigma \mathrm{/}{f}_0(500)$-meson pole, together with a background term, to lattice data at $m_{\pi }\approx 170\text{MeV}$. We find that the fitted residues are compatible with predictions from dilaton effective theory. In this framework, the $\sigma$-meson takes on the role of the dilaton, the Goldstone boson of spontaneously broken scale symmetry. These results support the idea that QCD may be governed by an infrared fixed point and offer a physical interpretation of the $D$-form factor (or $D$-term) in the soft limit.