Perturbative unitarity and the four-point vertices in the constructive standard model

We find a complete set of four-point vertices in the constructive Standard Model (CSM). This set is smaller than in Feynman diagrams as the CSM does not need or allow any additional four-point vertices (or “contact” terms) beyond what is present in Feynman diagrams and, furthermore, it does need or allow a four-point vertex for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Z</a:mi><a:mo>,</a:mo><a:mi>Z</a:mi><a:mo>,</a:mo><a:mover accent="true"><a:mi>W</a:mi><a:mo stretchy="false">¯</a:mo></a:mover><a:mo>,</a:mo><a:mi>W</a:mi></a:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>W</e:mi><e:mo>,</e:mo><e:mi>W</e:mi><e:mo>,</e:mo><e:mover accent="true"><e:mrow><e:mi>W</e:mi></e:mrow><e:mrow><e:mo stretchy="false">¯</e:mo></e:mrow></e:mover><e:mo>,</e:mo><e:mover accent="true"><e:mrow><e:mi>W</e:mi></e:mrow><e:mrow><e:mo stretchy="false">¯</e:mo></e:mrow></e:mover></e:mrow></e:math>, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>γ</k:mi><k:mo>,</k:mo><k:mi>Z</k:mi><k:mo>,</k:mo><k:mi>W</k:mi><k:mo>,</k:mo><k:mover accent="true"><k:mi>W</k:mi><k:mo stretchy="false">¯</k:mo></k:mover></k:math> or <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>γ</o:mi><o:mo>,</o:mo><o:mi>γ</o:mi><o:mo>,</o:mo><o:mi>W</o:mi><o:mo>,</o:mo><o:mover accent="true"><o:mi>W</o:mi><o:mo stretchy="false">¯</o:mo></o:mover></o:math>, in addition to the already known absence of the four-gluon vertex. We show that with this set of four-point vertices, perturbative unitarity is satisfied in the CSM. Additionally, we show that many constructive diagrams are not Feynman diagrams rewritten in spinor form. In fact, we show that there is a significant rearrangement of contributions from the diagrams in constructive calculations relative to Feynman diagrams for some processes. In addition to the already known or expected rearrangement in diagrams involving external photons, we also find that diagrams involving four-vector bosons are also significantly different from their Feynman counterparts.