Domain of existence and Standard Model obstruction for electroweak balls

Determine the boundary of the domain of existence of electroweak-ball solutions in the bosonic SU(2)×U(1) gauge-Higgs parameter space, and establish whether the failure to obtain solutions at the measured Standard Model couplings and Higgs-to-W mass ratio reflects genuine termination of the relevant solution families, limitations of numerical continuation, or restrictions imposed by the employed ansätze.

Background

The paper constructs spherical electric-type and axisymmetric magnetic-type electroweak balls for a range of the weak mixing parameter and Higgs self-coupling. The numerical families possess finite frequency intervals and two-branch structures, but the explored families do not reach the simultaneous measured Standard Model values of the gauge-coupling ratio and Higgs self-coupling ratio.

The authors emphasize that this failure is not a non-existence theorem. They lack an analytical characterization of the coupling-space boundary and cannot determine whether the apparent obstruction is physical, numerical, or a consequence of restricting attention to the specific electric- and magnetic-type ansätze. Characterizing this boundary would clarify whether electroweak balls can occur at the measured Standard Model point.

References

Within the families and parameter ranges investigated here, we have not found electroweak balls at the simultaneous measured Standard Model values of the couplings and mass ratios. The measured value of $\theta_{\rm W}$ can be approached for sufficiently small $\beta$ and large frequencies, whereas no solutions have so far been obtained at the measured Higgs-to-$W$ mass ratio. This observation should not be regarded as a non-existence result. At present there is no analytical characterization of the boundary of the domain of existence, and it remains unclear whether the obstruction reflects a genuine termination of the particular families studied here, a limitation of the numerical continuation, or the restricted nature of the Ans\

— Electroweak balls: non-topological solitons in the Weinberg-Salam theory  (2609.19293 - Herdeiro et al., 16 Sep 2026) in Section 4, Conclusions and outlook