Line operators in the gauged two-Higgs-doublet model

We investigate the global structure of the gauged two-higgs-doublet model (G2HDM), a framework that extends the Standard Model by introducing a dark sector governed by the gauge symmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msub><a:mo stretchy="false">)</a:mo><a:mi>X</a:mi></a:msub><a:mo>×</a:mo><a:mi>S</a:mi><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:msub><a:mo stretchy="false">)</a:mo><a:mi>H</a:mi></a:msub></a:math>. The full gauge symmetry of the theory, including the visible sector, is given by the universal covering group <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mover accent="true"><g:mi>G</g:mi><g:mo stretchy="false">˜</g:mo></g:mover><g:mo>=</g:mo><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>Y</g:mi></g:msub><g:mo>×</g:mo><g:mi>S</g:mi><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>L</g:mi></g:msub><g:mo>×</g:mo><g:mi>S</g:mi><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>3</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>C</g:mi></g:msub><g:mo>×</g:mo><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>X</g:mi></g:msub><g:mo>×</g:mo><g:mi>S</g:mi><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>H</g:mi></g:msub></g:math>. However, the true gauge group may instead be a quotient <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>G</u:mi><u:mo>=</u:mo><u:mover accent="true"><u:mi>G</u:mi><u:mo stretchy="false">˜</u:mo></u:mover><u:mo>/</u:mo><u:mi mathvariant="normal">Γ</u:mi></u:math>, where <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="normal">Γ</z:mi></z:math> is the center of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mover accent="true"><cb:mi>G</cb:mi><cb:mo stretchy="false">˜</cb:mo></cb:mover></cb:math> or a subgroup thereof, leading to different global structures that cannot be distinguished by local experiments. We explore the physical implications of these global structures, analyzing their effects on Wilson, ’t Hooft, and dyonic line operators, as well as the periodicity of the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>C</gb:mi><gb:mi>P</gb:mi></gb:math>-violating <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>θ</ib:mi></ib:math> angles associated with each group factor. Furthermore, we determine the minimal electric and magnetic charges that arise after electroweak symmetry breaking, highlighting their dependence on the choice of <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi mathvariant="normal">Γ</kb:mi></kb:math>. These findings provide a systematic characterization of the G2HDM’s global properties and their potential phenomenological consequences.