Dark symmetry implication for right-handed neutrinos

We argue that the long-standing issues of neutrino mass and dark matter can be manifestly solved in a dark gauge symmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msub><a:mo stretchy="false">)</a:mo><a:mi>D</a:mi></a:msub></a:math> that transforms nontrivially only for three right-handed neutrinos <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>ν</e:mi><e:mrow><e:mn>1</e:mn><e:mo>,</e:mo><e:mn>2</e:mn><e:mo>,</e:mo><e:mn>3</e:mn><e:mi>R</e:mi></e:mrow></e:msub></e:math>—the counterparts of known left-handed neutrinos. This theory assigns the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>ν</g:mi><g:mrow><g:mn>1</g:mn><g:mo>,</g:mo><g:mn>2</g:mn><g:mo>,</g:mo><g:mn>3</g:mn><g:mi>R</g:mi></g:mrow></g:msub></g:math> dark charge to be <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi><i:mo>=</i:mo><i:mn>0</i:mn></i:math>, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo>−</k:mo><k:mn>1</k:mn></k:math>, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo>+</m:mo><m:mn>1</m:mn></m:math>, respectively, in order for anomaly cancelation. Additionally, it imposes an inert Higgs doublet <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>η</o:mi></o:math> and two Higgs singlets <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>ξ</q:mi><q:mo>,</q:mo><q:mi>ϕ</q:mi></q:math> with dark charge <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>D</s:mi><s:mo>=</s:mo><s:mo>+</s:mo><s:mn>1</s:mn></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo>−</u:mo><u:mn>1</u:mn></u:math>, and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo>+</w:mo><w:mn>2</w:mn></w:math>, respectively. That said, the dark symmetry is broken by <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>ϕ</y:mi></y:math> (by two units) down to a dark parity <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>P</ab:mi><ab:mi>D</ab:mi></ab:msub><ab:mo>=</ab:mo><ab:mo stretchy="false">(</ab:mo><ab:mo>−</ab:mo><ab:mn>1</ab:mn><ab:msup><ab:mo stretchy="false">)</ab:mo><ab:mi>D</ab:mi></ab:msup></ab:math>, for which <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi>ν</eb:mi><eb:mrow><eb:mn>2</eb:mn><eb:mo>,</eb:mo><eb:mn>3</eb:mn><eb:mi>R</eb:mi></eb:mrow></eb:msub></eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mrow><gb:mi>η</gb:mi></gb:mrow></gb:math>, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mrow><ib:mi>ξ</ib:mi></ib:mrow></ib:math> are odd, whereas all other fields are even due to <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>D</kb:mi><kb:mo>=</kb:mo><kb:mn>0</kb:mn></kb:math>. The lightest of these odd fields is stabilized by <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>P</mb:mi><mb:mi>D</mb:mi></mb:msub></mb:math>, which is responsible for dark matter. Neutrino masses are generated by a scotoseesaw scheme, in which the seesaw part is mediated by <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msub><ob:mi>ν</ob:mi><ob:mrow><ob:mn>1</ob:mn><ob:mi>R</ob:mi></ob:mrow></ob:msub></ob:math>, while the scotogenic part is mediated by <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>ν</qb:mi><qb:mrow><qb:mn>2</qb:mn><qb:mo>,</qb:mo><qb:mn>3</qb:mn><qb:mi>R</qb:mi></qb:mrow></qb:msub></qb:math>, for which the hierarchy of atmospheric and solar neutrino mass splittings is explained.