Scalar and tensor meson dominance and gravitational form factors of the pion

We analyze the recent MIT lattice data for the gravitational form factors (GFFs) of the pion which extend up to $Q^2= 2~{\rm GeV}^2$ for $m_\pi=170$~MeV~\cite{Hackett:2023nkr}. We show that simple monopole fits comply with the old idea of meson dominance. We use Chiral Perturbation theory ($\chi$PT) to next-to-leading order (NLO) to transform the MIT data to the physical world with $m_\pi=140~$MeV and find that the spin-0 GFF is effectively saturated with the $f_0(600)$ and the spin-2 with the $f_2(1270)$, with monopole masses $m_\sigma= 630(60)$~MeV and $m_{f_2}= 1270(40)$~MeV. We determine in passing the chiral low energy constants (LECs) from the MIT lattice data alone \[ 10^3 \cdot L_{11} (m_\rho^2)=1.06(15) \, , \qquad 10^3 \cdot L_{12} (m_\rho^2)= -2.2(1) \, , \qquad 10^3 \cdot L_{13} (m_\rho^2) = -0.7(1.1). \] which agree in sign and order of magnitude % to be compared with the original estimates by Donoghue and Leutwyler. The corresponding D-term (druck) has the value $ D(0) = -0.95(3) $. We also analyze the sum rules based on perturbative QCD (pQCD) that imply that the corresponding spectral functions are not positive definite. We show that these sum rules are strongly violated in a variety of $\pi\pi-K \bar K$ coupled channel Omn\`es-Muskhelishvili calculations. This is not mended by the inclusion of the pQCD tail, suggesting the need for an extra negative spectral strength. Using a simple model implementing all sum rules, we find the expected onset of pQCD at very high momenta.