Quark masses and CKM hierarchies from $$S_4'$$ modular flavor symmetry

Abstract

Abstract We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $$S_4^\prime $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> <mml:mo>′</mml:mo> </mml:msubsup> </mml:math> modular flavor symmetry. The hierarchy is realized by the modulus $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> stabilized at $$\textrm{Im}\,\tau \gg 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Im</mml:mtext> <mml:mspace/> <mml:mi>τ</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , where the residual $$\mathbb {Z}_4^T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> symmetry is approximately unbroken and the Froggatt–Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa couplings. We also argue that $$\mathcal {O}\left( {0.1}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:mn>0.1</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> hierarchies of the $$\mathcal {O}\left( {1}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> coefficients and the spontaneous CP violation can be realized by imposing another $$S_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> modular symmetry.

Topics & Concepts