Nonholomorphic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> modular invariance for fermion masses and mixing in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> GUT

Abstract

Addressing the fermion flavor structures using modular invariance is a challenging task in the framework of quark-lepton unification. Building on recent applications of modular symmetry in nonsupersymmetric models, we propose the first renormalizable <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>5</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> grand unified theory incorporating level 3 nonholomorphic modular symmetry, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi mathvariant="normal">Γ</e:mi> <e:mn>3</e:mn> </e:msub> <e:mo>≃</e:mo> <e:msub> <e:mi>A</e:mi> <e:mn>4</e:mn> </e:msub> </e:math> . This framework constrains Yukawa couplings to polyharmonic Maaß forms, significantly reducing the number of free parameters while enhancing the predictive power of the models. We present a comprehensive analysis of fermion masses and mixing while tackling key grand unified theory (GUT) queries such as gauge coupling unification and proton decay. Beyond the minimal <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>S</h:mi> <h:mi>U</h:mi> <h:mo stretchy="false">(</h:mo> <h:mn>5</h:mn> <h:mo stretchy="false">)</h:mo> </h:math> framework, the Higgs sector incorporates the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mn>4</l:mn> <l:msub> <l:mn>5</l:mn> <l:mi>H</l:mi> </l:msub> </l:math> dimensional Higgs field crucial in differentiating the masses of down quarks and charged leptons, and the fermion sector is extended with three right-handed neutrinos enabling neutrino masses via the type-I seesaw mechanism. We analyze two benchmark models with distinct modular weight and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msub> <n:mi>A</n:mi> <n:mn>4</n:mn> </n:msub> </n:math> charge assignments. The predicted effective Majorana mass <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msub> <p:mi>m</p:mi> <p:mrow> <p:mi>β</p:mi> <p:mi>β</p:mi> </p:mrow> </p:msub> </p:math> values align with current neutrinoless double-beta decay experiments, and the effective neutrino mass <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>m</r:mi> <r:mi>β</r:mi> </r:msub> </r:math> is within the reach of future beta decay searches. The predicted sum of neutrino masses, <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mo>∑</t:mo> <t:msub> <t:mi>m</t:mi> <t:mi>i</t:mi> </t:msub> </t:math> , satisfies the upper bound set by recent cosmological observations. The gauge coupling unification is achieved through a light scalar triplet <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>ϕ</v:mi> <v:mn>3</v:mn> </v:msub> <v:mo>∼</v:mo> <v:mo stretchy="false">(</v:mo> <v:mn>3</v:mn> <v:mo>,</v:mo> <v:mn>3</v:mn> <v:mo>,</v:mo> <v:mo>−</v:mo> <v:mn>1</v:mn> <v:mo>/</v:mo> <v:mn>3</v:mn> <v:mo stretchy="false">)</v:mo> </v:math> and a scalar octet <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msub> <z:mi>ϕ</z:mi> <z:mn>5</z:mn> </z:msub> <z:mo>∼</z:mo> <z:mo stretchy="false">(</z:mo> <z:mn>8</z:mn> <z:mo>,</z:mo> <z:mn>2</z:mn> <z:mo>,</z:mo> <z:mn>1</z:mn> <z:mo>/</z:mo> <z:mn>2</z:mn> <z:mo stretchy="false">)</z:mo> </z:math> belonging to the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mn>4</db:mn> <db:msub> <db:mn>5</db:mn> <db:mi>H</db:mi> </db:msub> </db:math> Higgs, while proton decay constraints require that the contribution of the <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mn>4</fb:mn> <fb:msub> <fb:mn>5</fb:mn> <fb:mi>H</fb:mi> </fb:msub> </fb:math> Higgs to the up-quark mass matrix remains highly suppressed.

Topics & Concepts