Mechanical-radius inequality for hadrons

Determine whether the mechanical radius of the pion and the eta meson satisfies the conjectured inequality \(\langle r^2\rangle_{\mathrm{mech}} \geq -9D/(4m^2)\), where \(D=D(0)\) is the gravitational form factor at zero momentum transfer and \(m\) is the meson mass.

Background

The paper defines the matter radius from the slope of the gravitational form factor $A(t)$ at zero momentum transfer and the mechanical radius from the $D(t)$ gravitational form factor. The mechanical radius characterizes the spatial distribution of internal mechanical forces in a meson.

The authors explicitly identify the inequality $\langle r^2\rangle_{\mathrm{mech}} \geq -9D/(4m^2)$ as a conjecture and state that it will be tested using the pion and $\eta$-meson gravitational form factors calculated in the graviton soft-wall model.