<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Q</mml:mi></mml:math>-balls under a spontaneously broken U(1) symmetry

We study the evolution of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Q</a:mi></a:math>-balls under a spontaneously broken global U(1) symmetry. <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Q</c:mi></c:math>-balls are stabilized by the conservation of U(1) charge, but when the symmetry is spontaneously broken, the resulting Nambu-Goldstone boson can carry charge away from the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Q</e:mi></e:math>-ball, potentially leading to charge leakage. To study this process in a controlled setting, we consider a scenario where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>Q</g:mi></g:math>-balls first form under an unbroken U(1) symmetry, which is then spontaneously broken. We introduce two complex scalar fields: one responsible for forming the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>Q</i:mi></i:math>-ball, and the other for spontaneously breaking the U(1) symmetry, allowing us to clearly separate the formation and symmetry-breaking phases. Using numerical simulations in a spherically symmetric system, we find that the evolution of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>Q</k:mi></k:math>-balls depends sensitively on the structure of the interaction between the two fields and the magnitude of symmetry breaking. Depending on parameters, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>Q</m:mi></m:math>-balls can completely decay, evaporate into smaller, stable <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>Q</o:mi></o:math>-balls, or transition into oscillons/I-balls. In particular, we find that stable, localized remnants often survive the evolution over long timescales, especially when the symmetry-breaking scale is small. These results demonstrate that, even though spontaneous U(1) breaking can lead to significant energy and charge loss from <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>Q</q:mi></q:math>-balls, stable localized objects with reduced or no charge can frequently survive and potentially contribute to cosmological relics.