A Type-I seesaw framework with non-holomorphic modular symmetry

Abstract We study neutrino mass generation within the framework of non-holomorphic modular symmetry proposed by Qu and Ding. In this formalism, neutrino masses are generated via the Type-I seesaw mechanism, where the Yukawa couplings depend on non-holomorphic modular forms. The viability of the model is examined through a $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> analysis using current neutrino oscillation data. The $$\chi ^2_{min}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>min</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> value is found to be 7.06 for normal hierarchy(NH). All neutrino oscillation parameters are consistent within their $$1\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> allowed ranges, except the atmospheric mixing angle $$\sin ^2\theta _{23},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>23</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which is predicted to lie in the second octant. The Dirac CP-violating phase( $$\delta _{CP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>CP</mml:mi> </mml:mrow> </mml:msub> </mml:math> ) is constrained to the first and fourth quadrants, indicating relatively weak CP violation. These predictions can be tested in future long-baseline neutrino oscillation experiments. The sum of neutrino masses is compatible with the stringent bound proposed by the DESI experiment. However, the inverted hierarchy(IH) is not viable in this model, as the predicted value of $$\chi ^2_{min}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>min</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> exceeds 100, and the mixing angles $$\sin ^2\theta _{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>12</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $$\sin ^2\theta _{23}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>23</mml:mn> </mml:msub> </mml:mrow> </mml:math> lie outside the $$3 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> allowed ranges.