Texture zeros of quark mass matrices at fixed point $$\tau =\omega $$ in modular flavor symmetry
Shota Kikuchi, Tatsuo Kobayashi, Morimitsu Tanimoto, Hikaru Uchida
Abstract
Abstract We study systematically derivation of the specific texture zeros, that is the nearest neighbor interaction (NNI) form of the quark mass matrices at the fixed point $$\tau =\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point $$\tau =\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> in the $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> modular flavor symmetry by taking account multi-Higgs fields. Such texture zero structure originates from the ST charge of the residual symmetry $$Z_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> of SL (2, Z ). The NNI form can be realized at the fixed point $$\tau = \omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> in $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> and $$S_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> modular flavor models with two pairs of Higgs fields when we assign properly modular weights to Yukawa couplings and $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> and $$S_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> representations to three generations of quarks. We need four pairs of Higgs fields to realize the NNI form in $$A_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> modular flavor models.