Electroweak balls: non-topological solitons in the Weinberg-Salam theory

Abstract: We construct a new class of smooth, finite-energy solitons in the bosonic $SU(2)\times U(1)$ Weinberg-Salam theory, which we dub $electroweak$ $balls$. Their localization mechanism is analogous in spirit to that of $Q$-balls: the charged vector fields possess a harmonic time dependence while the energy-momentum tensor remains time independent. We explicitly construct both spherically symmetric electric-type solutions and axisymmetric magnetic-type solutions, and show that they form families characterized by a finite frequency interval, a mass gap, and a two-branch structure. These configurations provide electroweak counterparts of Proca-Higgs balls, with the vector-boson masses generated by the Higgs mechanism rather than introduced explicitly. The construction is not tied to the measured parameters of the Standard Model. More generally, it applies to bosonic electroweak-type sectors with different gauge couplings, Higgs self-coupling and symmetry-breaking scale, and hence potentially very different characteristic particle and soliton mass scales. For the families studied here, we do not find solutions at the measured Standard Model couplings and mass ratios.