We study the energy-momentum tensor of stable, metastable and unstable $Q$-balls in scalar field theories with U(1) symmetry. We calculate properties such as charge, mass, mean square radii and the constant $d_1$ (“$D$-term”) as functions of the phase space angular velocity $\omega$. We discuss the limits when $\omega$ approaches the boundaries of the region in which solutions exist, and derive analytical results for the quantities in these limits. The central result of this work is the rigorous proof that $d_1$ is strictly negative for all finite energy solutions in the $Q$-ball system.

• Received 25 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.076001