Strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> and flavor in multi-Higgs theories

We introduce a class of multi-Higgs doublet extensions of the Standard Model that solve the strong <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>C</a:mi><a:mi>P</a:mi></a:math> problem with profound consequences for the flavor sector. The Yukawa matrices are constrained to have many zero entries by a “Higgs-flavor” symmetry, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>G</c:mi><c:mi>HF</c:mi></c:msub></c:math>, that acts on Higgs and quark fields. The violation of both <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>C</e:mi><e:mi>P</e:mi></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>G</g:mi><g:mi>HF</g:mi></g:msub></g:math> occurs in the Higgs mass matrix so that, for certain choices of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>G</i:mi><i:mi>HF</i:mi></i:msub></i:math> charges, the strong <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>C</k:mi><k:mi>P</k:mi></k:math> parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mover accent="true"><m:mi>θ</m:mi><m:mo stretchy="false">¯</m:mo></m:mover></m:math> is zero at tree level. Radiative corrections to <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mover accent="true"><q:mi>θ</q:mi><q:mo stretchy="false">¯</q:mo></q:mover></q:math> are computed in this class of theories. They vanish in realistic two-Higgs doublet models with <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>G</u:mi><u:mi>HF</u:mi></u:msub><u:mo>=</u:mo><u:msub><u:mi mathvariant="double-struck">Z</u:mi><u:mn>3</u:mn></u:msub></u:math>. We also construct realistic three-Higgs models with <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mi>G</x:mi><x:mi>HF</x:mi></x:msub><x:mo>=</x:mo><x:mi mathvariant="normal">U</x:mi><x:mo stretchy="false">(</x:mo><x:mn>1</x:mn><x:mo stretchy="false">)</x:mo></x:math>, where the one-loop results for <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mover accent="true"><cb:mi>θ</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover></cb:math> are model-dependent. Requiring <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mover accent="true"><gb:mi>θ</gb:mi><gb:mo stretchy="false">¯</gb:mo></gb:mover><gb:mo>&lt;</gb:mo><gb:msup><gb:mn>10</gb:mn><gb:mrow><gb:mo>−</gb:mo><gb:mn>10</gb:mn></gb:mrow></gb:msup></gb:math> has important implications for the flavor problem by constraining the Yukawa coupling and Higgs mass matrices. Contributions to <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mover accent="true"><kb:mi>θ</kb:mi><kb:mo stretchy="false">¯</kb:mo></kb:mover></kb:math> from higher-dimension operators are computed at one loop and can also be sufficiently small, although the hierarchy problem of this class of theories is worse than in the Standard Model.