Transverse Densities of the~Energy-momentum Tensor and the~Gravitational Form Factors of the Pion

We present general features of the transverse densities of the stress-energy-momentum tensor \({\mit \Theta }^{\mu \nu }\) in the pion. We show positivity of the transverse density of \({\mit \Theta }^{++}\) (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 <span class="it">ia.</span> 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 \({\mit \Theta }^{\mu }_\mu (b)\) must change the sign, unlike the case of the positive definite \(J^+(b)\) or \({\mit \Theta }^{++}(b)\). 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. Abstract Published by the Jagiellonian University 2025 authors