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

TL;DR

The paper investigates gravitational form factors of the nucleon and pion using lattice data at $m_\\pi \approx 170$ MeV, fitting a $σ/f_0(500)$ pole plus a background to the $D(q^2)$ form factor to test an infrared conformal (dilaton) scenario for QCD. By comparing the Euclidean-residue fits to leading-order dilaton EFT predictions, the authors find that the nucleon residue $r_{E,σ}^N$ is consistent with the dilaton expectation $r_σ^N$, and the pion residue $r_{E,σ}^π$ aligns with the soft-pion constrained value $r_σ^π$, within uncertainties. The analysis suggests the σ-dominance in the nucleon D-term and yields a negative D-term in the infrared limit, supporting a physical interpretation of the D-form factor as tied to dilaton dynamics and spontaneous scale symmetry breaking. While the nucleon results robustly support the dilaton picture, the pion case remains less definitive for σ-dominance, highlighting the need for higher-precision data and extended kinematic coverage to sharpen conclusions about infrared fixed-point dynamics in QCD.

Abstract

We investigate the nucleon and pion gravitational $D$-form factors, by fitting a $σ/f_0(500)$-meson pole, together with a background term, to lattice data at $m_π\approx 170\text{MeV}$. We find that the fitted residues are compatible with predictions from dilaton effective theory. In this framework, the $σ$-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.

Figures (6)

• Figure 1: The $D$-form factors are fitted using a Euclidean pole parametrisation \ref{['eq:Nansatz']} and \ref{['eq:pionfunc']} with $r_{\mathrm{eff}} =0$ and $m_{\mathrm{E}, \sigma}=550\,\hbox{MeV}$. The fits are compared to the lattice data, shown in black, for the nucleon Hackett:2023rif (left) and for the pion Hackett:2023nkr (right). The dark curve indicates the central fit, while the shaded band represents the $68\%$ confidence interval.
• Figure 2: Diagrams for the self-energy corrections $\Sigma_{\lambda_3,g}$. The two diagrams for $\Sigma_{\lambda_4}$ are not shown.
• Figure 3: Feynman diagrams for the NLO corrections to the $\sigma NN$ vertex.
• Figure 4: The NLO form factor $F(q^2)$\ref{['eq:FNLO']} (left) and the corresponding density $\rho_F(s)=\tfrac{1}{\pi}\textrm{Im} F(s)$ (right). The input values \ref{['eq:free']} are $m_\pi=140\,\hbox{MeV}$, $F_\pi=93\,\hbox{MeV}$ and $\lambda=(g+1/2)^2$ for three different value of $\lambda$. The solid lines corresponds to the analytic linear $\sigma$-model computation and the dashed lines are fitted effective pole representations \ref{['eq:Feffective']}. The linear $\sigma$-model and effective pole curves are nearly identical in the Euclidean, despite rather different densities in the Minkowski region.
• Figure 5: The $D$-form factor of the nucleon (left) and pion (right) as a function of the momentum transfer $q^2$. Comparison between our main parametrisation (green) as also shown in Fig. \ref{['fig:170mevfits']}, i.e., Eqs. \ref{['eq:Nansatz']} and \ref{['eq:pionfunc']} with $r_{\mathrm{eff}} =0$, the $n$-pole fits performed in the original paper by Hackett et al. Hackett:2023rifHackett:2023nkr (blue) and in the meson dominance approach by Broniowski and Ruiz Arriola refs. Broniowski:2024oykBroniowski:2025ctl (red). The fits are compared to the data of refs. Hackett:2023rifHackett:2023nkr.