Gravitational form factors of pions, kaons and nucleons from dispersion relations

TL;DR

This work develops a model-independent dispersive framework, complemented by chiral perturbation theory, to determine the gravitational form factors of pions, kaons, and nucleons across energy scales. By incorporating modern $\pi\pi$ and $K\bar K$ scattering data via a coupled-channel Muskhelishvili-Omnès formalism and matching to ChPT, it delivers GFFs for pions and kaons, explores the pion-mass dependence up to $m_\pi=391$ MeV, and computes three-dimensional and two-dimensional density distributions inside the nucleon. The nucleon sector is extended with an NNLO matching to curved-spacetime ChPT, yielding $c_8=-4.28^{+0.37}_{-0.38}\ \mathrm{GeV}^{-1}$, $c_9=-0.68^{+0.06}_{-0.05}\ \mathrm{GeV}^{-1}$, and $\tilde d_{g4}=-0.04^{+0.01}_{-0.02}\ \mathrm{GeV}^{-3}$, providing a precise, testable benchmark for experiments and lattice QCD. The results illuminate the mass and mechanical distributions inside hadrons and establish a robust foundation for gravity-induced processes in low-energy QCD.

Abstract

The gravitational form factors of pions, kaons and the nucleons are investigated by employing modern dispersive techniques and chiral perturbation theory. We determine the gravitational form factors of pions and kaons, extending our analysis to explore the pion mass dependence of these form factors at several unphysical pion masses up to 391 MeV, for which lattice results exist for the meson-meson scattering phase shifts. We also review our analysis on the nucleon gravitational form factors at the physical pion mass, and then systematically calculate various three-dimensional spatial and two-dimensional transverse density distributions for the nucleons. These results provide new insights into the mass distribution inside nucleons. As a by-product, we match our dispersion relation results and those obtained from chiral perturbation theory with external gravitational source at the next-to-next-to-leading order, yielding values for the low-energy constants $c_8=-4.28_{-0.38}^{+0.37} ~\mathrm{GeV}^{-1}$ and $c_9=-0.68_{-0.05}^{+0.06} ~\mathrm{GeV}^{-1}$. These results offer a robust benchmark for future experimental and theoretical studies.

Figures (17)

• Figure 1: Elastic unitarity relation for the pion GFFs $F^\pi=\{A^\pi,D^\pi\}$Cao:2024zlf. The blue dashed lines denote the pions, the double wiggly lines represent the external QCD EMT current, and the dashed red vertical line indicates that the intermediate pions are on-shell.
• Figure 2: Results for real and imaginary parts of the components of the $D$-wave Omnès matrix.
• Figure 3: Results of the spectral function $\operatorname{Im}A^\pi(t)$.
• Figure 4: The total GFFs $A^{\pi/K}$ from single-channel Omnès solution Cao:2024zlf and coupled-channel MO solution. The blue dashed lines show the NLO ChPT prediction in the small-$|t|$ region. We also show the LQCD results at $m_\pi=170~$MeV in Ref. Hackett:2023nkr for comparison.
• Figure 5: Results of the spectral function $\operatorname{Im}\Theta^\pi(t)$. The blue dashed line shows the NLO ChPT prediction in small-$|t|$ region.