Transverse densities of the energy-momentum tensor and the gravitational form factors the pion

Abstract

We present general features of the transverse densities of the stress-energy-momentum tensor $\Theta^{\mu\nu}$ in the pion. We show positivity of the transverse density of $\Theta^{++}$ (analogous to the positivity of the transverse density of the electromagnetic current $J^+$) and discuss its consequences in conjunction with analyticity and quark-hadron duality, as well as the connection to $\pi\pi$ scattering at low energies. Our analysis takes into account the perturbative QCD effects, dominating at high momenta (or low transverse coordinate $b$), the effects of Chiral Perturbation Theory, dominating at low momenta (high $b$), and meson dominance in the intermediate region. We incorporate constraints form analyticity, leading to sum rules for the spectral densities of the corresponding form factors, which {\em i.a.} are relevant for the high-momentum (or the low-$b$) asymptotics. With the obtained high- and low-$b$ behavior, we deduce that the scalar (trace-anomaly) gravitational transverse density $\Theta^{\mu}_\mu(b)$ must change sign, unlike the case of the positive definite $J^+(b)$ or $\Theta^{++}(b)$. We also discuss the transverse pressure in the pion, which is positive and singular at low $b$, and negative at high $b$, in harmony with the stability criterion. The results for the form factors for space-like momenta are compared to the recent lattice QCD data.