<p>This research aims to address the limitations of the Standard Model (SM), which assumes massless neutrinos, by developing a model incorporating <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="201" /> </InlineMediaObject> <EquationSource Format="TEX">$SU(2)_L \times U(1)_Y \times T_7 \times Z_{10}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mi>L</mi> </msub> <mo>×</mo> <mi>U</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>Y</mi> </msub> <mo>×</mo> <msub> <mi>T</mi> <mn>7</mn> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mn>10</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> symmetry. Using a hybrid type I and type II seesaw mechanism, the model predicts neutrino masses and mixing parameters. Using Neural Network Algorithm (NNA), the study calculates neutrino mass eigenvalues as: <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="431" /> </InlineMediaObject> <EquationSource Format="TEX">$m_1=21.5411\ meV,\ m_2=23.2147\ meV,\ m_3=55.2442\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>21.5411</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> <mo>,</mo> <mspace width="4pt" /> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>23.2147</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> <mo>,</mo> <mspace width="4pt" /> <msub> <mi>m</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>55.2442</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="439" /> </InlineMediaObject> <EquationSource Format="TEX">$m_1=49.3498\ meV,\ m_2=50.1029\ meV,\ m_3=0.547342\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>49.3498</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> <mo>,</mo> <mspace width="4pt" /> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>50.1029</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> <mo>,</mo> <mspace width="4pt" /> <msub> <mi>m</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.547342</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation>), the matrix <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$U_{PMNS}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">PMNS</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and the effective neutrino mass parameters as: <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">$m_\beta =23.3706\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>β</mi> </msub> <mo>=</mo> <mn>23.3706</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">$m_{ee}=22.7984\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mrow> <mi mathvariant="italic">ee</mi> </mrow> </msub> <mo>=</mo> <mn>22.7984</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">$m_\beta =49.0997\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>β</mi> </msub> <mo>=</mo> <mn>49.0997</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">$m_{ee}=48.6039\ meV$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mrow> <mi mathvariant="italic">ee</mi> </mrow> </msub> <mo>=</mo> <mn>48.6039</mn> <mspace width="4pt" /> <mi>m</mi> <mi>e</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation>) for normal (inverted) mass hierarchy. The mixing angles (<InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta _{ab}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>θ</mi> <mrow> <mi mathvariant="italic">ab</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="164" /> </InlineMediaObject> <EquationSource Format="TEX">$ a<b\ \& \ a,b \in 1,\ 2,\ 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo><</mo> <mi>b</mi> <mspace width="4pt" /> <mo>&</mo> <mspace width="4pt" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt" /> <mn>2</mn> <mo>,</mo> <mspace width="4pt" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, Dirac <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and Majorana CP-violating phases, <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5993_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha , \ \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mspace width="4pt" /> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> are predicted as well. The predictions align well with the recent experimental data.</p>

Optimizing Neutrino Mass Predictions with Neural Network Algorithm and $SU(2)_L \times U(1)_Y \times T_7 \times Z_{10}$ Symmetry

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发表日期：

2025年4月29日

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Muhammad Waheed Aslam,
Abrar Ahmad Zafar

摘要

This research aims to address the limitations of the Standard Model (SM), which assumes massless neutrinos, by developing a model incorporating $SU(2)_L \times U(1)_Y \times T_7 \times Z_{10}$ $S U {(2)}_{L} \times U {(1)}_{Y} \times T_{7} \times Z_{10}$ symmetry. Using a hybrid type I and type II seesaw mechanism, the model predicts neutrino masses and mixing parameters. Using Neural Network Algorithm (NNA), the study calculates neutrino mass eigenvalues as: $m_1=21.5411\ meV,\ m_2=23.2147\ meV,\ m_3=55.2442\ meV$ $m_{1} = 21.5411 m e V, m_{2} = 23.2147 m e V, m_{3} = 55.2442 m e V$ ( $m_1=49.3498\ meV,\ m_2=50.1029\ meV,\ m_3=0.547342\ meV$ $m_{1} = 49.3498 m e V, m_{2} = 50.1029 m e V, m_{3} = 0.547342 m e V$ ), the matrix $U_{PMNS}$ $U_{PMNS}$ and the effective neutrino mass parameters as: $m_\beta =23.3706\ meV$ $m_{β} = 23.3706 m e V$ , $m_{ee}=22.7984\ meV$ $m_{ee} = 22.7984 m e V$ ( $m_\beta =49.0997\ meV$ $m_{β} = 49.0997 m e V$ , $m_{ee}=48.6039\ meV$ $m_{ee} = 48.6039 m e V$ ) for normal (inverted) mass hierarchy. The mixing angles ( $\theta _{ab}$ $θ_{ab}$ , with $ a<b\ \& \ a,b \in 1,\ 2,\ 3)$ $a < b & a, b \in 1, 2, 3)$ , Dirac $\delta $ $δ$ and Majorana CP-violating phases, $\alpha , \ \beta $ $α, β$ are predicted as well. The predictions align well with the recent experimental data.

Optimizing Neutrino Mass Predictions with Neural Network Algorithm and \(SU(2)_L \times U(1)_Y \times T_7 \times Z_{10}\) Symmetry

摘要