Q -balls and charged Q -balls in a two-scalar field theory with generalized Henon-Heiles potential

We construct <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Q</a:mi></a:math>-ball solutions from a model consisting of one massive scalar field <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ξ</c:mi></c:math> and one massive complex scalar field <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ϕ</e:mi></e:math> interacting via the cubic couplings <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msub><g:mrow><g:mi>g</g:mi></g:mrow><g:mrow><g:mn>1</g:mn></g:mrow></g:msub><g:mi>ξ</g:mi><g:msup><g:mrow><g:mi>ϕ</g:mi></g:mrow><g:mrow><g:mo>*</g:mo></g:mrow></g:msup><g:mi>ϕ</g:mi><g:mo>+</g:mo><g:msub><g:mrow><g:mi>g</g:mi></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msub><g:msup><g:mrow><g:mi>ξ</g:mi></g:mrow><g:mrow><g:mn>3</g:mn></g:mrow></g:msup></g:mrow></g:math>, typical of Henon-Heiles-like potentials. Although being formally simple, these couplings allow for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>Q</i:mi></i:math>-balls. In one spatial dimension, analytical solutions exist, either with vanishing or nonvanishing <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>ϕ</k:mi></k:math>. In three spatial dimensions, we numerically build <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>Q</m:mi></m:math>-ball solutions and investigate their behaviors when changing the relatives values of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>g</o:mi><o:mn>1</o:mn></o:msub></o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>g</q:mi><q:mn>2</q:mn></q:msub></q:math>. For <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>g</s:mi><s:mn>1</s:mn></s:msub><s:mo><</s:mo><s:msub><s:mi>g</s:mi><s:mn>2</s:mn></s:msub></s:math>, two <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>Q</u:mi></u:math>-balls with the same frequency exist, while <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>ω</w:mi><w:mo>=</w:mo><w:mn>0</w:mn></w:math> can be reached when <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>g</y:mi><y:mn>1</y:mn></y:msub><y:mo>></y:mo><y:msub><y:mi>g</y:mi><y:mn>2</y:mn></y:msub></y:math>. We then extend the former solutions by gauging the U(1) symmetry of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>ϕ</ab:mi></ab:math> and show that charged <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>Q</cb:mi></cb:math>-balls exist.