Pion and kaon D terms from holographic QCD and coupled-channel dispersion relations

Abstract

For a spin-zero meson, the exact trace identity relates the D term to the tensor gravitational form factor $A_M(t)$ and the total scalar trace ${\Theta }_M(t)$. We study the pion and kaon D terms by combining a holographic tensor form factor with a chiral-dispersive representation of the trace. Its normalization and slope at the origin are obtained from the forward Ward identity and curved-space SU(3) chiral perturbation theory. A two-channel $\pi \pi /K\bar{K}$ Muskhelishvili--Omnès solution constructed from empirical scattering amplitudes describes the continuation to spacelike momentum. Available lattice-QCD results are shown for comparison. In the chiral limit, saturation of the trace by the normalized explicit quark-mass response gives $D_{\pi }(0)=-1/3$, while the soft-pion theorem gives $D_{\pi }(0)=-1$. This comparison separates the explicit-mass contribution from the remaining chiral scalar response. At physical masses, SU(3) breaking in the chiral matching distinguishes the pion and kaon forward values, and coupled-channel rescattering governs their evolution away from the origin. The kaon D term is less negative than the pion result in the low-$Q^2$ spacelike region considered, with the separation decreasing as $Q^2$ increases. These form factors provide a low-energy reference for future lattice studies of the tensor and scalar channels.