Momentum-Current Gravitational Multipoles of Hadrons

Abstract

We study multipole expansion of the momentum currents in hadrons with three series ${S}^{(J)}$, ${\stackrel{\texttildelow{}}{T}}^{(J)}$, and ${T}^{(J)}$ in connection with the gravitational fields generated nearby. The momentum currents are related to their energy-momentum form factors, which in principle can be probed through processes like the deeply virtual Compton scattering currently studied at the Jefferson Lab 12 GeV facility and future Electron-Ion Collider. We define the leading momentum-current multipoles [the ``tensor monopole'' $\ensuremath{\tau}$ ($T0$) and ``scalar quadrupole'' ${\stackrel{^}{\ensuremath{\sigma}}}^{ij}$ ($S2$) moments], relating the former to the so-called $D$ term in the literature. We calculate the momentum-current distribution in the hydrogen atom and its monopole moment in the basic unit of ${\ensuremath{\tau}}_{0}={\ensuremath{\hbar}}^{2}/4{m}_{e}$, showing that the sign of the $D$ term has little to do with mechanical stability. The momentum-current distribution also strongly modifies the static gravitational field inside hadrons.