Extending the symmetries of the generalized <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>-symmetric 2HDM scalar potential to the Yukawa sector

There are only six independent types of symmetry-constrained (renormalizable) scalar potentials in the two Higgs doublet model (2HDM). For example, the scalar sector symmetry known as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi mathvariant="double-struck">Z</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msub><a:mo stretchy="false">⊗</a:mo><a:msub><a:mrow><a:mi mathvariant="normal">Π</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msub></a:mrow></a:math>, generated by the simultaneous application of two independent symmetries acting on the scalar fields, and the generalized <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>C</f:mi><f:mi>P</f:mi></f:math> symmetry known as GCP2 yield equivalent 2HDM scalar potentials. A similar situation arises for the scalar sector symmetries known as <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi mathvariant="normal">U</h:mi><h:mo stretchy="false">(</h:mo><h:mn>1</h:mn><h:mo stretchy="false">)</h:mo><h:mo stretchy="false">⊗</h:mo><h:msub><h:mi mathvariant="normal">Π</h:mi><h:mn>2</h:mn></h:msub></h:math> and GCP3, respectively. In this paper, we show that this “degeneracy” remains when the definitions of the corresponding symmetries are extended to the Yukawa sector with three quark generations. The proof involves the exploration of all possible extensions of the corresponding symmetries to the Yukawa sector, consistent with the phenomenological constraints of nonzero quark masses and a nontrivial quark mixing matrix. Moreover, we find that this result is a peculiarity of a Yukawa sector with three quark generations. In particular, with two quark generations, we find that models based on the extension of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi mathvariant="double-struck">Z</o:mi><o:mn>2</o:mn></o:msub><o:mo stretchy="false">⊗</o:mo><o:msub><o:mi mathvariant="normal">Π</o:mi><o:mn>2</o:mn></o:msub></o:math> to the Yukawa sector are inequivalent with those based on GCP2.